[KyleGoetz]

No. There are set verbs you use する to make. There are thousands of them, though. Think of it this way: really important verbs for things that have been around hundreds of years will have their own verbs. する verbs are often for highly technical verbs, newer words (テニス isn't exactly a thousand year old Japanese concept), and things like that.

But what I've just said is not a hard-and-fast rule. It's just something to show you that する verbage is very expansive.

Also, to sing = 歌う
to drink = 飲む
to eat = 食べる

Now, I mentioned する is often used for more technical/formal words. Here:
to sing an aria = 詠唱する
to imbibe/ingest [fancy words for eat/drink] = 摂取する