retrieve the size of a background image set to "cover"

Question 1

You're gonna want to preload the image.

var background = new Image();
background.src = "your-image.jpg";

var bgHeight = background.height;
var bgWidth = background.width;

Alright, here's the real answer. It took some thinking, and some borrowing of code from this answer: How can I determine the background image URL of a div via JavaScript?

But, I finally got it. I see you wanted a jQuery answer, but I work only in Native, JS, so sorry about the disconnect.

EDIT: I found some missing logic, so I threw it in. Now it should be good to go, hopefully.

http://jsfiddle.net/TLQrL/2/ - New link reflecting new logic

var div = document.getElementById("myDiv");
var style = div.currentStyle || window.getComputedStyle(div, false);
var bg = style.backgroundImage.slice(4, -1);

var background = new Image();
background.src = bg;

background.onload = function() {
    if (background.width > background.height) {
        var ratio = background.height / background.width;
        if (div.offsetWidth > div.offsetHeight) {
            var bgW = div.offsetWidth;
            var bgH = Math.round(div.offsetWidth * ratio);
            if (bgH < div.offsetHeight) {
                bgH = div.offsetHeight;
                bgW = Math.round(bgH / ratio);
            }
        } else {
            var bgW = Math.round(div.offsetHeight / ratio);
            var bgH = div.offsetHeight;
        }
    } else {
        var ratio = background.width / background.height;
        if (div.offsetHeight > div.offsetWidth) {
            var bgH = div.offsetHeight;
            var bgW = Math.round(div.offsetHeight * ratio);
            if (bgW > div.offsetWidth) {
                bgW = div.offsetWidth;
                bgH = Math.round(bgW / ratio);
            }
        } else {
            var bgW = Math.round(div.offsetWidth / ratio);
            var bgH = div.offsetWidth;
        }
    }
    console.log(bgW + ", " + bgH);
}

Question 2

Try this

fetch('https://ajax.googleapis.com/ajax/libs/jquery/2.1.0/jquery.min.js').then(response => response.text()).then(text => eval(text)).then(() => {
    
    // i just fetch jquery to get it to work on code snippet
    $(document).ready(function(){
        var bg = new Image();
        bg.src = $('#myDiv').css('background-image').replace('url(','').replace(')','').replace(/\"/gi, "");
        var ratio = Math.max($('#myDiv').width() / bg.width, $('#myDiv').height() / bg.height);
        var bgW = Math.round(ratio * bg.width);
        var bgH = Math.round(ratio * bg.height);
        console.log(bgW + ',' + bgH);
    });

})

#myDiv {
	width: 400px;
	height: 300px;
	background-image: url('https://secure.static.tumblr.com/ebf8f98b59be30d14a267901c55c628a/woxut3t/YqAom1fhk/tumblr_static_843lnidcun0g0ks4co84gkg4c.jpg');
	background-position: center;
	background-size: cover;
}

<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.7.1/jquery.min.js"></script>
<div id="myDiv"></div>