You can. Low-embedding degree may be bad due to the MOV attack, but pairing-friendly curves are particularly chosen so that the embedding degree is low but still enough to not decrease security. So any elliptic curve algorithm should be safe on the curve, not only pairing-based ones.
ECDSA if often used with NIST curves with cofactor 1. ...
The primitive you are looking at is called anonymous credentials. It deals exactly with your scenario, and there are dozen of papers on the subject. The introduction of this paper provides plenty of references to the literature on this subject.