To my knowledge, the only teleconverters that will work with the Pentax fit Sigma 70-200 HSMII are the Tamron 1.4x and the Kenko 1.5x DG. Both of those converters have additional contacts that connect to the HSM contacts on the lens, and AF is retained.
So, to try and answer your questions;
1. Yes
2. No!
3. Doesn't matter, as the Sigma's won't work anyway and therefore you have no choice.
I have the 70-200 HSMII and the Kenko 1.5x DG teleconverter, and I'm very pleased with the quality of the pictures. Here's one I posted on another thread;
The sigma converter probably should be better, but until they make one in Pentax fit that works with the Sigma HSM lenses, we won't know.