1. Is the 3-string braid group the only braid group isomorphic to a knot group?
Answer: no and yes. No for a trivial reason: the 2-string braid group is Z, the group of the unknot. If we take this trivial counter-example into account, then the answer is "yes", because the commutator subgroup of the braid group with more than 3 strings is finitely generated, but not free, while by Neuwirth-Stallings theorem the commutator subgroup of a classical knot group either is free and finitely generated or is not finitely generated.
2. Develop theory of fibered ribbon braids. Ask me what is a fibered ribbon braid.