The mechinism I have here is probably not correct. The actual mechanism may involve multiple steps/intermediates for example, but the idea is that Br is acting as a nucleophile and attacking the delta positive H atom attached to N, so it is a nucleophilic substitution (Br substituting H).

For the second question, B is the iodoform test, which works for most alcohols which have a methyl group attached to the carbon bonded to OH. It should be clear that phenol cannot give a positive result.