I just found that the answer to the first question about uniqueness is negative.
We have $f(34,29)=9$, and there are at least two essential different minimal tilings:
an irreducible one (squares of sides 19,15,14,10,10 clockwise and 4x1 in the middle)
a reducible one (squares of sides 17,17,12,12,6,4,4,4 clockwise and another one of side 6 in the middle)
A remark about coprime-reducible rectangles: given that the values $f(m,n)$ for given m and for coprime $n$ between $m/2$ and $2m$ are seemingly very close to each other as noted in the first thread, the value of a coprime-reducible one in this range would have to be about the double of the others. This sort of rules out their existence heuristically.