Finding the Shared Key In DHKE
You are confusing your notation. You are saying that $g^A\equiv 70 \bmod{71}$, but then you are also saying that $A\equiv 7^x\bmod{71}$. That is not true. So let's clarify the notation.

Let $A$ be Alice's private exponent and $B$ be Bob's private exponent.

You know that $B=28$ and you know that $g^A\equiv 70\bmod{71}$. You do not know $A$, however, but you do not need to know $A$.

The shared key between Alice and Bob is ${g^A}^B\equiv {g^B}^A \bmod{p}$. So given what you know, you can compute the shared secret.

To figure out what Alice's private exponent is, you would have to solve the discrete logarithm. For numbers this small, that would be easy, but in general (and for the size of numbers we use in practice) that is infeasible.