1 and 2. "If A then B" does NOT imply "if B then A".
in both A is "if I have my keys". That says nothing about "if I can open the door".
bi. An obvious way to prove that is to do the multiplications indicated on the right.
bii. You want to prove that n^4- n^2+ 1= 3k+ 1 for some number k. Subtracting 1 form both sides, n^4- n^2= 3k. Now look again at bi.