All of the evidence of magical events caused by Emma, I believe are correct. Somehow someway they involve her and some unexplained occurence (magic). Emma has to start believing first before she can use it. The first evidence of her believing it even if it is for a brief moment was when Jefferson told her, then when he dissappeared into the hat; only confirmed when she got a closer look at the hat. Nevertheless, I think she is still on the fence about magic but a lot closer to believing than when she first met Henry and continuing to the beginning of Hat Trick.
Magic, I haven’t thought that could be part of the request. Does anyone believe that, that was how Cora and Regina got magic. Emma, in Rumple’s case in regards to the deal if magic is it; she will be Rumple’s way of correcting or righting a past wrong and also so he can win?