For question 2, do you mean:
$$ax \ mod \ n = 1$$
$\forall x \in \mathbb{Z}$ ? (Note, no brackets, as me are talking about the operation)
(I suppose as long as $x$ is a number, it doesn't make much difference....)
For question 3:
$$XA^{n} = A^{n}x$$
What is $x$? It can't be a vector, because the above statement can't ever be true when $X, A \in \mathbb{M}_{2}$
Is $x$ a scalar ?