1) Science is capable of wonderful things - but not always, and rarely as quickly as we would wish.
This is a perfectly valid construction. Science has the ability to do/create/etc. wonderful things.
2) Science has capability of wonderful things - but not always, and rarely as quickly as we would wish.
This is nonsense. Adding 'the' before 'capability', as others have mentioned, actually doesn't help; the sentence is still nonsense. "Science has the capability of wonderful things" doesn't make any sense, because there's no verb here; science has the capability of doing what to/with/for/etc. wonderful things? Creating them, making them, doing them, destroying them...what? This is not valid.
3) Science has capability of doing wonderful things - but not always, and rarely as quickly as we would wish.
Here you've added the verb 'doing', which is great. We do need 'the' here, though. I'd also mention that, while in the first example science is capable of something, here it must have the capability to do something. The correct version would be:
Science has the capability to do wonderful things - but not always, and rarely as quickly as we would wish.
This has identical meaning to the original sentence #1, just with a lot more words. To be capable of something is the same thing as having the capability to do something.
4) Science has capability of making wonderful things - but not always, and rarely as quickly as we would wish.
Let's fix this one up similarly to sentence #3, and for the same reasons:
Science has the capability to make wonderful things - but not always, and rarely as quickly as we would wish.
This is similar in meaning to the first sentence and corrected third sentence, except it uses make where the third sentence uses do (and the first sentence leaves out the verb entirely).
You can always add the verb to the first sentence, by the way; any of these are valid:
Science is capable of (doing/making/creating/etc.) wonderful things - but not always, and rarely as quickly as we would wish.
To answer your final question, a variant of the first sentence is most likely your best choice. But you can modify it with any verb to add the specificity it seems you were trying to add in sentences three and four.