I am a tad confused with what you are trying to show here. I cannot see where the k fits in...

I interpreted it as: if x\in A\<h> then x \not\equiv h^n mod N for all n (x lies in a different coset of A/N form any power of h).

However, this is not true. Take A=C_2xC_2, and let N=<(2, 2)>. Take H=<(2, 0)> and let x=(0, 2). Then (0, 2)=(2, 0) mod N...so I think I have interpreted this incorrectly...but I can't see any other way of interpreting it...

For your second question, and abelian group will do (as then x^{-1}h^{-1}xh=1). The example I gave above works here.