I've been trying to add a background image to a <div> on one of my pages. I'm aware of the CSS way of doing so, but Fandom combines HTML and CSS properties into one thing, such as background-color, etc. Is background-image a compatible Fandom attribute? I've been having trouble with getting it to work.
CSS and HTML are meant to work together. There is no such thing as a pure CSS site or a pure HTML site. Putting CSS in HTML tags is not a "Fandom thing", it is called inline styling and it is part of official HTML specifications.
The example you have given uses purely inline styling. Due to the markup interpreter (which is a "Fandom thing" since it needs to handle both MediaWiki markup and Fandom's custom tags), you cannot specify a background image using inline styling (note that there are also limits on the HTML tags you can use). However, as Fngplg was trying to say, it can be done using site-wide (i.e. stylesheet) CSS. You can make stylesheets the same way you make any other page on your wiki; just be sure the page name ends in ".css". Note, however, that only a few pre-named stylesheets are actually loaded. For more information on these pre-named stylesheets, see Help:Community_CSS_and_JS, Help:Advanced_CSS_and_JS and Help:Personal_CSS_and_JS.
You can use additional stylesheets through use of @import. For example, I have created my own stylesheet to fix some display issues in Internet Explorer 11 (here). I then include it in my global (personal) CSS. As such, that CSS is applied when I view any Wikia site.
Could you link to the page you are testing it on? I will need to check to see if any other CSS is overriding it.
Alternatively, you can check yourself by using the development tools of any common desktop/laptop web browser. The most common way of accessing them is right-clicking on the page component you wish to examine and then selecting "Inspect Element" (or some equivalent) from the context menu provided by the browser.