A8 is a fairly large group. It works on 8 objects (1,2,3,4,5,6,7,8), but it has more elements (permutations). I think it has 8!/2 elements, so the largest order is much larger than 8.
1. What is the order of (123)? Apply this 3 times and you get the identity, so the order is 3. So what is the order of the element (45678)? And what is the order of (123)(45678)?
2. The order of (1234) is 4 and 4 is not divisible by 3. (Verify this suggestion with somebody else).
3. In S4 you have 4 objects (1,2,3,4) to work with. What is the order of the element (1234)? How about the group formed by the 2 elements (12) and (34)? Is the latter group formed by either element by itself? If not, is it cyclic?