Following up on this thread in case anyone else is looking for something similar.
Turns out upgrading to PHP 5 was easy, so I tried cformsII, which seemed to have an impressive set of features. I couldn’t get it working, though, and the plugin’s site was not helpful.
Anyway, I ended up going with Contact Form 7 together with Contact Form to DB Extension. This setup gave me all I wanted: multiple forms, trigger emails for user and admin, storage of submitted forms in database, and data exports. It took a little more time to set up my form, but the plugin and extension both worked out of the box, which is a huge plus for me.