The standard other exemptions are doctor-patient, attorney-client, religious leader-religious person, and spousal confidentiality laws. Some, but not all, states also shield journalist's sources. In the last case, in states without a shield law, going to jail to protect a source's identity is considered a professional honor.
Yes. I mean, that's the whole point of that part of the fifth amendment. There would be no need for an explicit carveout of self-incrimination if the government didn't have the general ability to compel witnesses to testify.