As I said before, I've never written a GAP program and still haven't. I think what you mean is you want to write a loop that does all the calculations. In GAP, the fundamental data structure is the list. So for your first question, you need to construct the list L in a loop (I guessed that m=(p^4-1)/2). I haven't even downloaded GAP, but from what I read, the following is close to what you want:
You can try the above code and see what happens. I really don't want to get into GAP programming.
L:=[ ]; #empty list
while i<=m do
Add(L,[i,LogFFE(z^i + 1,z)]; # i.e., j=LogFFE(z^i+1,z) is the power with z^j=z^i +1
L; # display the list L