Full Viewport Height Scaling Div Just CSS No Js... Possible

Maintain aspect ratio of div but fill screen width and height in CSS?

There is now a new CSS property specified to address this: object-fit.

https://developer.mozilla.org/en-US/docs/Web/CSS/object-fit

The feature is widely supported by now (http://caniuse.com/#feat=object-fit).

Div height 100% and expands to fit content

Here is what you should do in the CSS style, on the main div

display: block;
overflow: auto;

And do not touch height

height of a div to match the height of the viewport

Yes, it's possible. CSS percentage height will work as long as the parent element has a height defined. Assuming this is your markup:

<html>
  <body>
    <div></div>
  </body>
</html>

This CSS will cause the <div> to have the full height of the viewport:

html, body, div {
  height: 100%;
}

If you wanted the <div> to expand with its content, the CSS would change as follows:

html, body {
  height: 100%;
}

div {
  min-height: 100%;
}

Scale div with its content to fit window

let outer = document.getElementById('outer'),
        wrapper = document.getElementById('wrap'),
        maxWidth  = outer.clientWidth,
        maxHeight = outer.clientHeight;
window.addEventListener("resize", resize);
resize();
function resize(){let scale,
    width = window.innerWidth,
  height = window.innerHeight,
  isMax = width >= maxWidth && height >= maxHeight;

    scale = Math.min(width/maxWidth, height/maxHeight);
    outer.style.transform = isMax?'':'scale(' + scale + ')';
    wrapper.style.width = isMax?'':maxWidth * scale;
    wrapper.style.height = isMax?'':maxHeight * scale;
}

body {
  font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, "Helvetica Neue", Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji";
  background: #e6e9f0;
  margin: 10px 0;
}

#wrap {
  position: relative;
  width: 640px;
  height: 480px;
  margin: 0 auto;
}
#outer {
  position: relative;
  width: 640px;
  height: 280px;
  background: url('https://source.unsplash.com/random') no-repeat center center;
  transform-origin: 0% 50%;
  border-radius: 10px;
  box-shadow: 0px 3px 25px rgba(0, 0, 0, 0.2);
  overflow: hidden;
}
#outer:before {
  content: "";
  position: absolute;
  bottom: 0;
  width: 100%;
  height: 100px;
  -webkit-backdrop-filter: blur(20px);
  backdrop-filter: blur(20px);
}

#profile {
  background: url('https://source.unsplash.com/random/300x300') no-repeat center center;
  position: absolute;
  width: 60px;
  height: 60px;
  bottom: 0;
  margin: 20px;
  border-radius: 100px;
  background-size: contain;
}
#content {
  font-size: 20px;
  position: absolute;
  left: 0px;
  bottom: 0;
  margin: 30px 100px;
  color: white;
  text-shadow: 0 1px 2px rgba(0,0,0,0.5);
}

#content div:last-child {
  font-size: 15px;
  opacity: 0.7;
}

<div id="wrap">
<div id="outer">
  <div id="profile"></div>
  <div id="content">
    <div>Monwell Partee</div>
    <div>UX / UI Designer</div>
  </div>
</div>
</div>

How can an html element fill out 100% of the remaining screen height, using css only?

The trick to this is specifying 100% height on the html and body elements.
Some browsers look to the parent elements (html, body) to calculate the height.

<html>
    <body>
        <div id="Header">
        </div>
        <div id="Content">
        </div>
    </body>
</html>

html, body
{
    height: 100%;
}
#Header
{
    width: 960px;
    height: 150px;
}
#Content
{
    height: 100%;
    width: 960px;
}

Font scaling based on size of container

If the container is not the body, CSS Tricks covers all of your options in Fitting Text to a Container.

If the container is the body, what you are looking for is Viewport-percentage lengths:

The viewport-percentage lengths are relative to the size of the initial containing block. When the height or width of the initial containing block is changed, they are scaled accordingly. However, when the value of overflow on the root element is auto, any scroll bars are assumed not to exist.

The values are:

vw (% of the viewport width)
vh (% of the viewport height)
vi (1% of the viewport size in the direction of the root element's inline axis)
vb (1% of the viewport size in the direction of the root element's block axis)
vmin (the smaller of vw or vh)
vmax (the larger or vw or vh)

1 v* is equal to 1% of the initial containing block.

Using it looks like this:

p {
    font-size: 4vw;
}

As you can see, when the viewport width increases, so do the font-size, without needing to use media queries.

These values are a sizing unit, just like px or em, so they can be used to size other elements as well, such as width, margin, or padding.

Browser support is pretty good, but you'll likely need a fallback, such as:

p {
    font-size: 16px;
    font-size: 4vw;
}

Check out the support statistics: http://caniuse.com/#feat=viewport-units.

Also, check out CSS-Tricks for a broader look: Viewport Sized Typography

Here's a nice article about setting minimum/maximum sizes and exercising a bit more control over the sizes: Precise control over responsive typography

And here's an article about setting your size using calc() so that the text fills the viewport: http://codepen.io/CrocoDillon/pen/fBJxu

Also, please view this article, which uses a technique dubbed 'molten leading' to adjust the line-height as well. Molten Leading in CSS