Finding a primitive root

For your first question, consider $p=71$ and $g=11$. Here, $11$ is a primitive root modulo $71$, but $11^{70}\equiv 1 \bmod 71^2$. Unfortunately, this is not an example of your second question, because $7\bmod 71$ is also a primitive root.

I found this example looking at lists of Wieferich primes with respect to a certain base. There is probably an example for your second question farther down the list; I see no reason why there couldn't be one.

I just checked the first $103$ odd primes $p$ and none of them satisfy $(2)$, with $g$ the smallest positive primitive root modulo $p$.

With computer search I found that $g=5$ for $p = 40487$ is an example of (2) . http://www.wolframalpha.com/input/?i=5%5E40486+mod+1639197169

I would guess that this should happen with probability $1/p$. $\sum_{2 < p \le 40487} \frac 1p = 2.1235$ only. In retrospect it makes sense that the examples are rare.