Sorry, Vec(G) is the category of G-graded vector spaces. And yes, my claim is that if C and D are Morita equivalent fusion categories, then the corresponding Turaev-Viro TQFTs are equivalent. Unfortunately, I don't have a good reference. It is probably hidden somewhere in one of Kevin Walker's papers though. It is much easier to justify if you think about TQFT from a physicists perspective

Yes, this is true. For example, Rep(G) and Vec(G) define the same TQFT. If two categories are 2-Morita equivalent, they have the same associated TQFTs