Would that be feasible to "trial and error" h by running a few tries and choose the best with respect to some known derivative on fixed point x0?

I would pick an h where objects don't displace by more than a fraction of their size each frame.

How could we pick the right h?

Is there a general agreed upon method to pick h?

I agree with the method of sampling h. Can do so at high granularity and decrease if necessary