In case anyone with the same problem comes across this thread, I figured out the problem. Very simple, stupid stuff.
Problem 1: When I reinstalled Apache, the /etc/apache2/sites-available/000-default.conf file reverted the DocumentRoot to /var/www, when it was supposed to be /var/www/html.
Problem 2: There was an additional and nearly identical .htaccess file in /var/www/html/omeka that had a typo. It was missing the ^ before install/.*$
Also, the IT department who set up the cloud server for me installed 2 instances of Omeka (who knows why), one under /var/www and one under /var/www/html, so this whole problem most likely would have been averted (or at least easier to solve) if that wasn't the case.