I prefer your first proof. It gives the reason why the remainder is 0, 1 or 4, as opposed to listing all the squares, which gives you no real mathematical insight.

For the second question, note that you cannot possibly write 7 as a sum of three terms chosen (possibly with repetitions) from the set {0,1,4} (doing it mod 8). Thus the remainder must be 0, 1, 2, 3, 4, 5 or 6.