is theadditivegroup of integers mod n. So the element 1 has order n (that's the number of times you mustadd1 to itself in order to get a multiple of n). Also, the order of 0 is 1. So the first of those four choices is the correct one.

That is not true. Euler's theorem tells you that . But this number is not unique. For example, if n=10 and a=3 then , but also .

I never heard of Carmichael's indicator (nor did Google, as far as I can tell).