can $\frac{1}{H(z)}$ be causal and stable?
For 1/H(z) to be causal, it must not have more zeroes than poles. Therefore, H(z) must not have more poles than zeroes.

If H(z) has linear phase, then it must have zeroes either in the unitary circle or simultaneously inside and outside of it. Therefore, 1/H(z) can't be stable if H(z) has linear phase.