Coherence for symmetric monoidal categories

Solution 1:

1. Yes. By functoriality and strong monoidality, $F\gamma$ must satisfy the symmetry axioms.
2. This is precisely the statement of coherence for SMC's. Mac Lane gives the braided version at the end of XI.5. Note, however, that this does not imply that "all" diagrams of the correct type commute. Only those corresponding to the same element of the symmetric group.
3. The fact that you say "the" symmetry map corresponding the given permutation is already a consequence of coherence for SMC's. All symmetry maps corresponding to the given permutation (including $\gamma_{a,b}$) are equal.
4. Nope, c.f. #2. In fact, the "all diagrams commute"-type coherence theorem is quite atypical. Most monoidal categories with extra structure have a coherence theorem more along the lines of "this is how to (efficiently) decide which diagrams commute: (...)".