"Do I really have to recompile PHP each time I want to add a new module?"
Yup.
"Are these all the modules I have installed"
Correct.
"My biggest worry is that if I recompile PHP I might leave something important out such as one of the modules I already have installed"
Copy the 'configure' command from the PHPinfo screen and add your own requirements to it.
It's worth a bit of time to check just what is installed, and wether you actually need everything that's installed.
Also, it helps if you make a script to recompile PHP. You're bound to make a few mistakes the first few times, and doing everything by hand once is cute, twice is not so cute, and by then 15th time you're completely fed up with it. 🙂
A forum, a FAQ, email notification, what else do you need?