How to find a small generator with small inverse? Does it have negative impact to security? (for Schnorr subgroup of $\mathbb{Z}/P\mathbb{Z}$)
Q3.) Would a small $g, g^{-1}$ have any impact to security?


It would not appear so.  It is easy to show that, if we had a generator $g$ that makes the Discrete Log (or the Computational Diffie-Hellman) problem easy, then the Discrete Log/CDH problem is easy with respect to any base $h$ that's within the same subgroup.

Suppose we had a way that, given $g, g^x$, gave us the value $x$.  Then, what we could do, given $h, h^y$ is first find the value $z$ such that $g^z = h$ (by solving the discrete log to the base $g$), and then the value $w$ such that $g^w = h^y$ (again, solving another discrete log to the base $g$); then, we have $h^{wz^{-1}} = h^y$; since we know $w$ and $z$, that tells us the discrete log.  Similar (but more complicated) logic allows us to solve the Computational Diffie-Hellman problem, given a CDH oracle with respect to base $g$.

However, this is not a complete answer to the question; it might be that the existence of a small $g, g^{-1}$ makes all discrete log/CDH problems easy.

I don't know of a way this can be disproven.  However:


None of the known discrete log methods have significant speed-ups when given a small $g, g^{-1}$ (by significant, I mean more than speeding the operation of multiplying by $g$)
For just about any prime, there will be tiny $g$ values that generate groups that contain the Schnorr group as a subgroup.  If there were a fast way to complete discrete logs with a tiny $g$ in this group, then you could use that to compute discrete logs in the Schnorr group (and although this argument doesn't take into account the tiny $g^{-1}$, there's no obvious way to use that).