First one:

You were on the right track. Fermat's theorem is the way to go. We know that 2^52 = 1 (mod 53). 155 = 51 + 2*52 (as you mentioned) so 2^155 = 2^51 (mod 53), and 2^51 = (2^52)/2... So 2^51 = (2^52)/2 (mod 53) = 1/2 (mod 53) = 54/2 (mod 53) (since 54 = 1 (mod 53)) = 27 (mod 53).

Second one:

Sorry I'm lost.

Third one:

Order is the same as the number of elements in a group. So you should find a subgroup with 4 elements. If you're given a generator, g, of the original group (the one with 16 elements, of order 16), a typical generator for a subset of order 4 is g^4... Since (g^4)^4 = g^16 = 1, the identity so your subgroup would be g^4, (g^4)^2 = g^8, (g^4)^3 = g^12, (g^4)^4 = g^16 = 1.