- I regret damaging your book.
That is a perfectly good sentence.
- I regret that I've damaged your book.
That one is also good. (The first one is, I think, more likely. Also more likely is: "I'm sorry I damaged your book.") You could also say: "I regret having damaged your book."
- I regret to damage your book.
- I regret to have damaged your book.
Neither of those sentences is English in structure.
(Say: "Neither seems right" or "Both are wrong", but you can't use "both" in the negative (with "not").)
- I have no reason for regret.
No, I'm not sorry--not yet.