I assume you mean F* is the set of non-zero elements of F. Don't you know the multiplicative group of F is cyclic? So this is really a question about finite cyclic groups.
1. If the order is 2n-1, then if x is a generator of F*, so also is x2.
2. If p is odd, consider the homomorphism which sends every element to its square. The image is then the set of squares; thus you should be able to figure out the order of this image from the fundamental theorem of homomorphisms.