The popup slide is illegal to breakup the double play. Do I call a second out at first even if I do not believe the batter runner would have been out had there not been an illegal slide?

Yes, assuming there was contact (2-32-2a "...pop-up slide into the fielder")

I don't think contact is required. An illegal slide is illegal.

And when is a pop-up slide illegal? All the time? Or only when there's contact?

When done to break up a double play, it would be illegal, regardless of whether contact was made. It would be like not sliding or avoiding when running to 2B, by popping up you have done the same thing, placed your body in the way. Let's say 2B is taking the throw from SS on the back side of the bag, and the runner pops up in his face, but does not make contact - FPSR violation, 2 outs.

If there is no play at the bag he is sliding to, a pop up would be legal.