Part 2: Define Y = Xb. . Remember that when we write down , t is just some argument. If we wanted to evaluate , we would just use the definition of the MGF to end up with .
Part 3: Do the same thing basically, remembering that, with respect to the expectation operator, everything other than X is a constant, and can be moved in an out with the usual rules. You should also use this method to redo part 1, since your method of just replacing X with whatever function of X you want hasn't exactly been justified.