1. If you can calculate rough bounds for the terms and their derivatives in the winding number integral, you can figure out how densely you need to sample the curve to calculate the integral exactly.

2. If the winding number is zero, there still might be a root in the region because poles and zeros contribute opposite winding number, and they might exactly cancel.

3. Polynomials are great because the bounds are easy to compute and because you don't have to worry about cancelation since the only pole is at infinity.

You'd think so, but check out the 'smooth surprise' in Kahan's great writeup: https://people.eecs.berkeley.edu/~wkahan/Mindless.pdf

We can approximate quite a bit, but ultimately further analysis is required to show that there are no such surprises on any particular root-finding problem.