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When do I need to create a new variable within a function?

I'm a new learner of JavaScript, and when I get to learn the way of using a function. It sometime confuses me on why we should declare a new variable and add the variable to the action we want to execute. Let's look into the code.
function reverse(word){
Array.from(word);
let reverseWord='';
for(i = word.length-1; i >= 0; i--) {
reverseWord += word[i];
}
return reverseWord;
}
I'm sure you know this one of the way of reversing string in javascript, my question is:
Why do we need to declare a new variable within the function, when should we declare it?
Why can't I just type console.log(word[i]);?
What does it mean by wordLength+=word[i];?
Why should we return the new variable(wordLength), instead of the function(reverse) after the loop?

Why do we need to declare a new variable within the function...
Because you need a place to store the reversed word as you build it. (Note: wordLength isn't a good name for that variable. It doesn't contain the word's length. It contains the characters of the reversed word.)
...when should we declare it?
Any time before you first need it.
Why can't I just type console.log(word[i]);?
Because the goal of the exercise is to build a string containing the reversed word, not just to output it. (And because console.log writes a new line each time you call it.)
What does it mean by wordLength+=word[i];?
That adds the character in word[i] to the end of wordLength. For instance, if the word is "start", wordLength starts out with "", then gets "t" added to it to make it "t", then gets "r" added to it to make "tr", and so on.
(+= is a shorthand way to write wordLength = wordLength + word[i];. There are several of these compound assignment operators, most of them for math: -=, *=, etc.)
Side note: The Array.from call in your code isn't doing anything useful. It's creating an array, but then throwing that array away because nothing uses the return value. The rest of the code is using the string you receive in word.

Why do we need to declare a new variable within the function, when should we declare it?
Vars is a place to store data. If your algorithm requires keeping some data to use it later you need vars. Also well named variables is a good way to create easy-to-understand code
Why can't I just type console.log(word[i]);?
You can, but it will do nothing useful. Your goal is to build a string and return it. Usage of your function will be something like
const word = getSomeText()
const reversedText = reverse(word)
doSomeStuff(reversedText) // whatever, send it online, or render it on screen some fancy way, not in the console.
So you need to return actual string, not to solve a puzzle and show the answer whatever way you like
Why should we return the new variable(wordLength), instead of the function(reverse) after the loop?
Because it contains reversed word and you function supposed to return it. there is rare complicated occasions when a function returning itself is useful, but it has nothing in common with your task

Why do we need to declare a new variable within the function, when should we declare it?
Variable is required to store the data value that can be changed later on. In your case wordLength variable is required to store the reverse string.
It's best to declare variables when you first use them to ensure that they are always initialized to some valid value.
Why can't I just type console.log(word[i])
console.log() is used just to print the output but will not use if you want to return something and as per your statement it will just print the word[i] not a whole reverse string.
What does it mean by wordLength+=word[i]
It means you are concatenating the each iteration word[i] into a wordLength variable.
wordLength+=word[i] is a shorthand for wordLength = wordLength + word[i]. If the left hand side of the + operator is a string, JavaScript will coerce the right hand side to a string.
Why should we return the new variable(wordLength), instead of the function(reverse) after the loop ?
Because this is what you expected from the function. It returns the reversed string and function should return it.

Javascript: Delete operator

I am looking at the Mozilla Developers website on the concept of the delete operator. In the last sub section of the page referring to “Deleting array elements” two similar scripts are shown, but the only difference in the scripts is how they modified the array.
In the first script, I quite don’t understand why “if” statement does not run. My current understanding is that delete operator “removes the element of the array”. If I were to type trees[3] in the console, it would return undefined in the console.
var trees = ["redwood","bay","cedar","oak","maple"];
delete trees[3];
if (3 in trees) {
// this does not get executed
}
The second script seems to "mimic" the delete, but not literally. Undefined is assigned to trees[3]. It doesn’t make sense to me how the “if” block runs in this script, but the first example does not. Can anyone help me understand this JavaScript behavior?
var trees = ["redwood","bay","cedar","oak","maple"];
trees[3] = undefined;
if (3 in trees) {
// this gets executed
}

There is a huge difference between the two methods you are trying:
Method 1:
You are deleting, destroying, completely removing the key 3 in your array called tree, hence there is no 3 in tree left, and the if check returns false.
Method 2:
You are assigning a new value to the key 3, which is undefined, there is still 3 in tree, and the if check returns true.

In your second example the key 3 still exists. It just holds a value that happens to be undefined. It IS confusing, but that's just the way Javascript is.
The in operator just checks if the key exists, not if the value is defined.
If you were to output the whole arrays after each of your "deletions" the first example would display something like this:
["redwood", "bay", "cedar", 4: "maple"]
Whilst the second example would print out something like this:
["redwood", "bay", "cedar", undefined, "maple"]
So as you can see, in your first example the key is completely missing and it continues with the next key which is 4. In the second example the key still exists, but it's value is set to undefined.

There is a difference between undefined which is set by the user and undefined which the javascript engine returns once something is actually undefined, meaning doesn't exist.
javascript can tell the difference between the two.
So in your example, when you do this:
var trees = ["redwood","bay","cedar","oak","maple"];
trees[3] = undefined;
if (3 in trees) {
console.log("hi");
}
javascript can tell that property 3 exists, but it was set to undefined by the user.
to prove so you have the following:
if (5 in trees) {
console.log("hi");
}
the property 5 of the array was never created, javascript knows it's undefined
by lack of creation and regards it as a property which doesn't exist, and therefore doesn't display the "hi"

if(3 in tree) {
//Stuff that won't get executed
}
is in fact correct, the thing is that in operator in Javascript does not work like in python, it simply checks if an object has a proprety. An array in javascript has a proprety 0, just like a string has a proprety 2 with the value someString[2].
the difference between delete object[prop]; and object[prop] = undefined; can be seen through object.hasOwnProperty(prop); or iterating through values or props of the object.

Javascript for loop syntax

As javascript developers we all have to write a lot of for loops. Before a couple of months I saw an alternative syntax, which I really liked. However, I'm now interested, is there any other nice way.
Let's say that I have an array of data representing users in a system. What I did before is:
var users = [
{ name: "A"},
{ name: "B"},
{ name: "C"},
{ name: "D"},
{ name: "E"}
];
var numOfUsers = users.length;
for(var i=0; i<numOfUsers; i++) {
var user = users[i];
// ...
}
There is one additional row var user = users[i];. Normally I feel more comfortable if I have user instead of users[i]. So, the new way:
for(var i=0; user=users[i]; i++) {
// ...
}
I'm also wondering if the second approach produces problems in some of the browsers. One of my colleagues reported that this syntax is a little bit buggy under IE.
Edit:
Thankfully, the answers below pointed me out to the right direction. If some of the elements of the array is falsy then the loop will stop. There is some kind of solution:
for(var i=0; typeof (user=users[i]) !== "undefined"; i++) {
// ...
}
But that's too much for me. So, I guess that I'll use this syntax only when I'm 100% sure that all the elements are truly (which means never :)).

In your “new” approach, you don’t need numOfUsers any more.
As for the potential problems: This approach relies on all users[i] having values evaluating to true for the loop to continue (and user becoming undefined, equal to false and therefor ending the loop after the last user is processed) – but sometimes you might have data where not every record evaluates to true, but “false-y” values might also occur in the data – and in that case, this approach of course fails.

The problem with this approach:
for(var i=0; user=users[i]; i++) {
// ...
}
...is that it assumes user won't be "falsey" (0, "", null, undefined, NaN, or of course false) until you've gone past the end of the array. So it'll work well with an array of non-null object references, but if you then get in the habit of using it, it will bite you when you have an array of numbers, or strings, or such.
The other reason not to declare variables within the for construct is that it's misleading: Those variables are not scoped to the for loop, they're function-wide. (JavaScript's var doesn't have block scope, only function or global scope; ES6 will get let which will have block scope.)
On modern JavaScript engines (or with an "ES5 shim"), you can of course do this:
users.forEach(function(user) {
// ...
});
...which has the advantage of brevity and not having to declare i or numUsers or even user (since it's an argument to the iteration callback, and nicely scoped to that). If you're worried about the runtime cost of doing a function call for each entry, don't be. It'll be washed out by whatever actual work you're doing in the function.

I'm amazed if the second syntax works at all your middle operation should evaluate to true for each loop you want to complete and false as soon as you want to be done looping. As for any issues with your first for loop, a JavaScript is function scoped so that inner var statement will still leak to the containing function (as well as that i). This is different than most other languages that have block scoping. It's not so much of a problem but something to keep in mind if you are debugging.

If you are already using jQuery, you can use the jQuery.each function to loop over your arrays.
In any case you can look at the source code of that function and copy the relevant parts for your own foreach function: http://james.padolsey.com/jquery/#v=1.10.2&fn=jQuery.each

Javascript: Calling Object Properties Multiple Times

I have an object:
var object = {
string1 : 'hello',
string2 : 'world'
}
And If I want to call the string2 property of the object is it slower to call it multiple times like:
...object.string2...
...object.string2...
or it would be faster to create a reference for it what holds the value of the parameter like:
var string2 = object.string2;
...string2...
...string2...
The reason why I think that the second one could be faster, because I think right now that the first one always scans the whole object to grab the value.

You are correct - the second one is faster, because JavaScript does not need to perform the lookup of string2 each time. The change is even more profound in something like this:
(do stuff with foo.bar.baz.qux)
versus
var property = foo.bar.baz.qux;
(do stuff with property)
In that example, foo must be scanned for bar. Then bar must be scanned for baz. Et cetera.
In your example the gain will be minimal unless you are doing a lot of work with string2, but you are correct in saying that it is faster.

In you case this does not matter . But for big object with big prototype chunks - you right.
But you may win in speed but loose in functionality because var a = obje.property coping by value - not by reference and if obj.property will be change dynamically that variable a will have old value of obj.property

javascript ternary operator inside variable

okay im working with a friend and he sent me js file which included a variable that included the ternary operator. I cant figure out how to change it to if..else. can you help please?
also i noticed ".length" didnt have the normal "()" after it, is there a reason why?
var nextRadioTab = activeRadioTab.next().length ? activeRadioTab.next() : $('#contentslider div:eq(0)');

Does this work?
if (activeRadioTab.next().length) {
var nextRadioTab = activeRadioTab.next();
} else {
var nextRadioTab = $('#contentslider div:eq(0)');
}
In JavaScript, objects are more-or-less just a list of names pointing to values. Each name-value pair is called a "property".
These values themselves can be any type of value, including a function. If the value of a property is a function, we call that a "method".
Say you want an object to track the x and y coordinates of a point.
var point = { x: 10, y: 20 };
In this case we can just use simple values, because we don't need any behaviour more advanced than getting a value (alert(point.x)) or setting one (point.x = 10).
jQuery is designed to let your code work on different browsers; different browsers behave differently in lots of situations, so jQuery can't just let you set
element.text = "hello world"
because depending on the type of object element is, it will need to modify different properties on different browsers. For this reason, jQuery makes you use methods for things like this:
element.text("hello world")
The .length attribute of a jQuery object is simple; it's controlled by jQuery itself and doesn't need to do any special things in different browsers. For this reason, you just use it directly. If they needed more complicated behaviour, they would use a function/method instead:
var myObject = { length: 2 }; // myObject.length
var myObject = { length: function() { return 2; } }; // myObject.length()

var nextRadioTab;
if (activeRadioTab.next().length)
nextRadioTab = activeRadioTab.next();
else
nextRadioTab = $('#contentslider div:eq(0)');
length is a property of whatever next() returns, which is most likely the same type of object as activeRadioTab.