RE: Is there any reason I couldn’t use DIVI Builder to create nice content around the Connections plugin on certain pages?
Not exactly sure what you are asking… you can use Divi as you normally would to build you page. You would but the Connections shortcode in one of its content blocks to embed the directory.
RE: Is there a transparent background option on the default template? … can you increase the font size of listings?
Custom CSS can be used to make transparent backgrounds and change font sizes. In many cases the font sizes are simply inherited from the theme, but not all.
Hope this helps!