Yes, there are many constructions for computing over private data that use homomorphic encryption. For example, Paillier's cryptosystem, which supports addition

Yes, there are many constructions for computing over private data that use homomorphic encryption. For example, Paillier's cryptosystem, which supports addition of ciphertexts, has been a building block in many voting protocols. You can also do things like with homomorphic encryption like private set intersections -- meaning that two people could see what's in the intersection of their private sets without revealing anything outside the intersection. Gentry's work is still a long way away from being practical, but is an exciting theoretical result nonetheless.