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Why are my HTML elements moving around depending on screen size or zoom?

My HTML and CSS elements are moving around depending on the screen size or zoom of a page. I have screenshots of this example.
This is what it ends up looking like
What it SHOULD look like

If your HTML and CSS elements change with the screen size you can either write media queries or try putting your CSS in terms of percentages rather than pixels. For instance given this CSS code:
.some-class {
width: 300px;
padding: 25px;
margin-top: 10px;
}
All of those measurements in px are going to look very different on a phone screen vs a laptop screen because 300px is about 1/2 the width of a phone screen but only about 30% of a laptop screen. Rather than specify width in px, we can say we want some-class to take up a certain percentage of the screen like this:
.some-class{
width: 30%;
}
Or like this:
.some-class{
width: 30vw;
}
For the padding and margin we can write a media query, which changes the CSS according to screen size. Assuming our original code is for a laptop and we want to make it fit on a phone screen we can add the following to our CSS stylesheet:
#media screen and (max-width: 480px;){
.some-class {
padding: 15px;
margin-top: 5px;
}
}
This overwrites the original CSS if it detects a screen smaller than 480px across. It's worth knowing how to write media queries anyway in case something that is horizontally aligned on a laptop should be vertically aligned on a phone so here's a helpful link.
https://www.w3schools.com/Css/css3_mediaqueries_ex.asp

How to scale the viewport to a minimum width

Problem: I have a page that looks terrible <480px and doesn't display enough relevant information to the user.
Attempted Solutions:
(ex. 320px screen) set the initial scale to 1.5, but then I need to set the scale accordingly for all the screen sizes between 320-480px.
(ex. 320px screen) set the width of your viewport to 480px, however this makes you need to scroll around the screen instead of zooming out like setting the scale would do.
Question: What it seems I need is a combination of the two solutions. One that will scale my viewport, but only until it shows a min-width such as 480px worth of content on the screen. Is this possible without javascript or is solution #1 what I would need to do?
Other considerations: Solution needs to work on all browsers/mobile (IE11+)

I'm not 100% sure what you are trying to do but if I understand correctly, you can set this in css. Setting the width to 100% will keep it flexible to your viewport window & setting a minimum width will not allow it it get any smaller than that.
html,body {
width: 100%;
min-width: 480px;
height: auto;
}

Have you attempted to use media queries ?
For example:
#media screen and (min-width: 480px) {
body {
background-color: lightgreen;
}
}

Dynamic margins using css

I'm quite new to web-design, but know most of the basic stuff, but please try to keep it relatively simple) So, I'm designing a web-site, where I want to have grey background on the sides of the main page, which I managed to do. However, I want the page to "eat up" the grey space if the window is resized, just like Marriott hotel uses I have tried looking their code up, but it's a lot and I don't think they use CSS but JavaScript for resizing dynamics. Am I right? If so, is there a way of doing it with CSS? Or java, but relatively simple, I'm very new to it!
Also, I have currently given the wrapper a width of 88% and used the 12% as the grey background. Is using % the best solution for different screen resolutions? (Keeping in mind that it would be for PCs and tablets at the most, no smartphones)

Using percentages can be useful, but if you truly want things to look how you want them to on a smaller screen resolution, you can use media queries.
Media queries work well because they can even change the styles depending on device orientation:
#media (min-width: 700px) and (orientation: landscape) { ... }
Using a media query. (Make the window bigger & smaller to see it in action.)

I think what you want is the following:
Add a background to the body.
Add a div with X px and margin auto, that will center it.
So then, when you resize the window, this will "eat" the margins from the body hidding the grey background.
If what you want is different sizes for the web depending on the width screen, then you are entering to media queries.

You are looking for:
#wrapper { width: 900px; margin: 0 auto; }
This will place your wrapper (with a 900px width) with even margins on the side and then recalculate the margins as the window changes.
I would give your wrapper a set width since setting a width of 88% means that the center content will ALWAYS be 88% of the window which means you will always have margins.

You may do it this way
here is the fiddle: http://jsfiddle.net/fcvbkv5w/
body {
text-align: center;
}
#navbar {
width: 100%;
margin-left:0%;
background-color: black;
text-align: center;
}
#fwcenter {
width: 70%;
position: relative;
text-align: center;
background-color: red;
}
#media (max-width: 700px){
#fwcenter{
width:100%;
}
}
<div id="navbar">
<center>
<div id="fwcenter">
<div id="container">
Website content
</div>
</div>
</center>
</div>

Image no longer responsive as put in table cell layout

I'm trying to center an image on a site. As I want it to be centered both horizontally and vertically I used a table/table-cell layout as following:
<div id="mainContainer>
<div id="horizon">
<img src="url">
</div>
</div>
Here mainContainer ist set to display: table and horizon is set to display: table-cell.
Unfortunately the image is no longer resizing proportionally as it becomes part of this structure. As soon as I move it into the table/table-cell divs it resizes to its original size (instead of being resized proportionally because of max-width: 100% and max-height: 100%)
see: http://jsfiddle.net/U8KcN/
EDIT:
My bad. I just tried to simplify the issue. I want to build a little slideshow for images. The problem here is that I do not know which sizes the images are (referring to someone else using the slideshow). So in simple terms I need a specific CSS that 1) centers the image vertically and horizontally in the div if its width and height are smaller than the div's dimensions, 2) resizes the image automatically if it is bigger than the div's dimensions (unnecessary to add, it is unknown whether it's width or height is bigger).
source: OP Comment

Every time I hear someone talk about the evils of using tables and then see them create a complete table structure out of DIVs with pretty much the same amount of markup -- sometimes more -- it really gets on my nerves.
By the way, have you ever developed a site for someone who wanted to do some of their own layouts in the content area using a CMS like Wordpress? The only way someone with almost no HTML knowledge can do that well is by allowing them to use table tools in Tiny MCE or another editor. So yes, tables are still important.
Recently, I had to do just that for a customer, but the images in the table cells were not responding and shrinking for smaller screen sizes. All I had to add to my CSS to care of that problem was:
table {
table-layout: fixed;
}
Now the images in the table cells are growing and shrinking accordingly, and the horizontal scrollbar is no longer appearing. Oh, of course you have to use percentages in your table width and td widths as well.

How about dropping that "CSS-table" stuff and doing it a bit easier?
<div style="width:auto;height:auto;margin:25%;text-align:center;vertical-align:middle">
<img src="URL">
</div>
At least, that’s how I would handle it...
EDIT:
Please note that I've put the CSS inline to show you what element should get what style. In production, you should — as a comment to this answer correctly stated — always separate style from code like. So, practically, you'll end up with something like this:
<style>
.centerimg {
width:auto;
height:auto;
margin:25%;
text-align:center;
vertical-align:middle
}
</style>
...
<div class="centerimg">
<img src="#">
</div>
EDIT 2:
Replying to the related comment, here's the update to make the image fit it's parent proportionally:
If your image has a bigger width than height...
<style>
...
img{
max-width:100%;
height:auto
}
</style>
or, if your image has a smaller width than height...
<style>
...
img{
max-height:100%;
width:auto
}
</style>
EDIT 3:
Looking at your fiddle, I came up with this which works like you want it to work:
<style>
*{
width:100%;
height:100%;
margin:0;
text-align:center;
vertical-align:middle
}
img{
width:auto;
height:100%;
}
</style>
I've forked your fiddle to show the updates: http://jsfiddle.net/LPrkb/1/
EDIT 3:
As the OP doesn't seem to be able to decide what he needs, I'm adding this final edit due to his latest comment.
You could also use a CSS background-image with "background-size:contain" on the "mainContainer" and be done with it... check http://jsfiddle.net/HGpfJ/2/ or look at this 100% working example taking a completely different approach which results in just the same effect/functionality:
<!DOCTYPE HTML>
<html itemscope="itemscope" itemtype="http://schema.org/WebPage">
<head>
<title>Example</title>
<style>
html,body{width:100%;height:100%;margin:0;padding:0}
#centerimg{
width:100%;
height:100%;
background:transparent url(http://oi42.tinypic.com/v9g8i.jpg) no-repeat scroll center;
background-size:contain;
}
</style>
<body>
<div id="centerimg"></div>
</body>
</html>
Let's face the facts: depending on where in the document structure you want to have the image "centered", there are more than a dozen of ways to do it.
If OP needs specific code, we will need the complete document structure from OP and not simply a "generalized" code-snippet which could be anywhere in whatever document structure.

Using tables (display: table-cell) for layout is a web development anti-pattern these days. Try using the <span> element with display: inline-block to vertically and horizontally center the image, as this method will work all the way back to IE 6. Also, you can change the image size to be a percentage if you want it to resize according to its container:
http://jsfiddle.net/hHWy8/1/
HTML:
<span class="horizontal">
<span class="vertical">
<img src="url" />
</span>
</span>
CSS:
span {
display: inline-block;
}
span.horizontal {
text-align: center;
width: 100%;
}
span.vertical {
vertical-align: middle;
border: 1px solid black; /* put a border on container for display purposes */
line-height: 1000px; /* this sets the vertical height */
overflow: hidden;
white-space: nowrap;
}
span.vertical img {
height: 50px; /* set your image height (could be percent) */
width: 50px; /* set your image width (could be percent) */
}
span.vertical br {
display: none;
}

To resize your image you could use this css:
html,body,div,img{
height:100%;
}
img { width:100%}
It sets the html and body's height to 100%, this is needed so the height of your page takes all available space

How do I expand selector width from center (or other efficient alternatives)?

I've got a navigational bar (#nav_bar), which has the following CSS properties:
#nav_bar {
background: url(../images/navbar.png) repeat-x;
width: 100%; height: 50px;
margin-bottom: 20px;
}
It takes the width of #wrap which is 1024px wide and has margin: auto;, however I would like to expand it so that it will fit all screen sizes 100%. I attempted to set width: 500%; just to see what it would do, then I realized that it expands from the left -> right, rather than both ways from the center.
So, what I'm asking is;
Is it possible to have an element expand from the center, then
perhaps I could set the max-width property or use javascript to
find out the visitors screen resolution then assign the width from
there; without major inefficiencies, i.e. extended load times/cross-browser compatibility issues?
Just for reference, a link to the particular page I'm talking about
Any answers will be greatly appreciated ;)!

Simply move your #nav_bar out of the #wrap.
Alternatively you can make your #nav_bar have position: absolute; left: 0px; width: 100%; in CSS, that will work too.

Why don't you use CSS3 Media Queries, to find out about screen size of your clients.
If your #nav-bar is a block-level element, like a div, a ul or a p element, then it by default would take the whole width of its container. Thus you don't need to set width: 100%; there. Also, you can use text-align: center; to center align the content.
In your case, you can use absolute positioning with overflow: visible attribute, and set the width of the menu. Also, you may simply extract your #nav-bar out of the wrap, to let it take the whole space.

use margin: auto
you can see an example here: http://jsfiddle.net/s995c/4/