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How can I remove an element that is NOT on the DOM using jQuery?

I want to use jQuery to manipulate a cloned element that is not on the DOM, to perform actions like .remove() on it. Say I have the following code:
var div= $('<div> <div id="div1"></div> </div>');
div.remove('#div1');
console.log(div.html());
The result on the console will still show that the element was not removed. string manipulation is not desirable, I'm looking for something analogue to $().remove()

The div variable will contain a reference to the outer div. You need to use find() to get the inner div by its id:
var $div = $('<div><div id="div1"></div></div>');
$div.find('#div1').remove();

Using the context argument of the jQuery() function:
$('div', div).remove('#div1');

How to replace all id from href?

I have the following markup
<div class = "general">
<div id ="custom"></div>
</div>
How to change id = "custom" in all <div> with class="general" from href on page using jQuery?

You can try this:
$("div.general").each(function() {
$(this).children("div#custom").text($(this).children("a").attr("href"));
});
If I understand you correctly, you want to iterate through all div.generals, and change the text of each child div#custom to the href of the child a.
See a working example on JSfiddle.
Also, another tip is to avoid using multiple elements with the same id. In your code you have a <div> with id="custom". You also say that the div.general appears multiple times — therefore, the id "custom" will appear multiple times. This is bad practice. I suggest that you change id to class.

You need to loop through all div.general and replace the id attribute of div#custom to whatever is there as the anchors href property. The following code will work:
$(".general").each(function(){
$(this).find("#custom").attr("id", $(this).find("a").attr("href").replace("#", ""));
})
Here the .find() will dig out elements from any depth inside the parent. If you are sure about the DOM position of the elements, you can change the .find() to .children()

Add class to parent div

I am working with the google maps drawing manager. They don't put id's or class names on the drawing tools button bar so I'm trying to do this myself.
First I want to remove the circle button which the below works fine, but I want to add my own button so need to add a class name to the parent div "gmnoprint" but google has about 5 div's all with the same class name. I just want to add it to the one where the circle button was found.
<div class=gmnoprint"></div>
<div class=gmnoprint"></div>
<div class=gmnoprint"></div>
<div class=gmnoprint">
<div>
<div> <== This is what I found in my search
<span>
<div>
<img></img>
</div>
</span>
</div>
</div>
</div>
I am able to find the element I want and remove it, but adding a class to its wrapper div is proving a bit difficult for me.
This works for removing the button
$(".gmnoprint").each(function(){
$(this).find("[title='Draw a circle']").remove();
});
This doesn't work.. Just add's the class to all ".gmnoprint" div's
$(".gmnoprint").each(function(){
$(this).find("[title='Draw a circle']").remove().parent().addClass("test");
});

remove() removes the element from the DOM and returns the free-standing jquery object which has no connection to the DOM at all. A call to parent() after calling remove() is incorrect and that likely is the cause for your issue.
Try splitting your statements to:
var toRemove = $(this).find("[title='Draw a circle']");
toRemove.parent().addClass("test");
toRemove.remove();

You can use jQuery insertAfter and out your button after that default button then remove it.
$(".gmnoprint").each(function(){
var defBtn = $(this).find("[title='Draw a circle']");
$('<button class="my-button" />').insertAfter(defBtn);
defBtn.remove();
});
Or use jQuery after like this:
$(".gmnoprint").each(function(){
$(this)
.find("[title='Draw a circle']")
.after($('<button class="my-button" />'))
.remove();
});

You can use child selector to target the elements
$(".gmnoprint > div > div").addClass('myClassName');
At that point you could replace the html of the whole div , or find the span and replace it's inner html. Using html() method you don't need to use remove() as it will replace all contents of the element(s)
$(".gmnoprint > div > div").addClass('myClassName').find('span').html('<newButton>');
API Reference : http://api.jquery.com/child-selector/

How do I insert a block of html after a block that I have just rendered on the DOM

I am building a block of nested divs around a specific element on the client. I am using jQuery's .wrap() initially to get the first part of the block which is working fine. But now I want to attach the ending block after the element I am wrapping and I am finding I can't attach it to anything because it is all being created at the same time. I tried using insertAfter() but I don't want it to be a sibling of the element I am wrapping, I want it to come after it's parent.
Here is my code:
$(document).ready(function() {
buildShadows('#section');
});
function buildShadows(element){
$(element).wrap("<div class='section_shadow_wrapper'><div class='section_shadow_curve'><div class='section_shadow_outer'><div class='section_shadow_inner_left'><div class='section_shadow_inner_right'>");
$("<div class='section_shadow_bottom_left'><div class='section_test_bottom_right'>").insertAfter(element);
}
What I am trying to do is append the first element of the second part (section_shadow_bottom_left) as a sibling of 'section_shadow_inner_right' but after 'element'
Any help would be appreciated.
Thanks.

You should be able to just traverse up to the new parent you just created.
Have you tried this?
function buildShadows(element){
$(element).wrap('<div class="section_shadow_wrapper clearfix"><div class="section_shadow_curve"><div class="section_shadow_outer"><div class="section_shadow_inner_left"><div class="section_shadow_inner_right">')
.parent().after('<div class="section_shadow_bottom_left"><div class="section_test_bottom_right">');
}

Try traversing to the next sibling of the original element and using .insertBefore() on it.
var nextsibling = $(element).next();
//Wrap code
$("<div class='section_shadow_bottom_left'><div class='section_test_bottom_right'>").insertBefore(nextsibling);

How to clear all <div>s’ contents inside a parent <div>?

I have a div <div id="masterdiv"> which has several child <div>s.
Example:
<div id="masterdiv">
<div id="childdiv1" />
<div id="childdiv2" />
<div id="childdiv3" />
</div>
How to clear the contents of all child <div>s inside the master <div> using jQuery?

jQuery's empty() function does just that:
$('#masterdiv').empty();
clears the master div.
$('#masterdiv div').empty();
clears all the child divs, but leaves the master intact.

jQuery('#masterdiv div').html('');

Use jQuery's CSS selector syntax to select all div elements inside the element with id masterdiv. Then call empty() to clear the contents.
$('#masterdiv div').empty();
Using text('') or html('') will cause some string parsing to take place, which generally is a bad idea when working with the DOM. Try and use DOM manipulation methods that do not involve string representations of DOM objects wherever possible.

I know this is a jQuery related question, but I believe someone might get here expecting a pure Javascript solution. So, if you were trying to do this using js, you could use the innerHTML property and set it to an empty string.
document.getElementById('masterdiv').innerHTML = '';

jQuery recommend you use ".empty()",".remove()",".detach()"
if you needed delete all element in element, use this code :
$('#target_id').empty();
if you needed delete all element, Use this code:
$('#target_id').remove();
i and jQuery group not recommend for use SET FUNCTION like .html() .attr() .text() , what is that? it's IF YOU WANT TO SET ANYTHING YOU NEED
ref :https://learn.jquery.com/using-jquery-core/manipulating-elements/

If all the divs inside that masterdiv needs to be cleared, it this.
$('#masterdiv div').html('');
else, you need to iterate on all the div children of #masterdiv, and check if the id starts with childdiv.
$('#masterdiv div').each(
function(element){
if(element.attr('id').substr(0, 8) == "childdiv")
{
element.html('');
}
}
);

The better way is :
$( ".masterdiv" ).empty();

$("#masterdiv div").text("");

$("#masterdiv > *").text("")
or
$("#masterdiv").children().text("")

$('#div_id').empty();
or
$('.div_class').empty();
Works Fine to remove contents inside a div

You can use .empty() function to clear all the child elements
$(document).ready(function () {
$("#button").click(function () {
//only the content inside of the element will be deleted
$("#masterdiv").empty();
});
});
To see the comparison between jquery .empty(), .hide(), .remove() and .detach() follow here http://www.voidtricks.com/jquery-empty-hide-remove-detach/

When you are appending data into div by id using any service or database, first try it empty, like this:
var json = jsonParse(data.d);
$('#divname').empty();

$("#masterdiv div[id^='childdiv']").each(function(el){$(el).empty();});
or
$("#masterdiv").find("div[id^='childdiv']").each(function(el){$(el).empty();});

try them if it help.
$('.div_parent .div_child').empty();
$('#div_parent #div_child').empty();