Does RSA-OEAP have integrity and authenticity properties? RSA-OAEP is IND-CCA2 secure (indistinguishable under an adaptive chosen ciphertext attack). Does it also have the INT-CTXT (integrity of ciphertext) and INT-PTXT (integrity of plaintext) properties?

I don't believe that IND-CCA2 implies these. Are they true for RSA-OAEP though?
INT-CTXT and INT-PTXT are usually on considered for private-key encryption.  For public-key encryption, no correct encryption scheme can satisfy those requirements.  (Proof:  The adversary can run the encryption algorithm on an arbitrary message and submit it as its output.  Since it made no queries to its encryption oracle, this ciphertext violates both INT-CTXT and INT-PTXT security.)