Is it okay to pigtail 12AWG and 14AWG on a 15AMP circuit?
Yes.
Yes. It's no problem as long as you can get a good twist on the wires.
Pig tails can be 14 gauge since the branch circuit is protected by a 15A Breaker.

You'll end up with four wires in the wire nuts, 1 12AWG and 3 14AWG, so make sure you use the right size wire nut.
True, never more than one wire per screw. Lots of people don't like the back stabbing... Can you blame them? The name even sounds wrong. And wrapping clockwise generally improves the tightening operation if there is a slight twist in the wire but there is an argument that a pull on a clockwise hook puts a counter clockwise torque on the screw. I've heard arguments for both but I say either is fine if you tighten the screw to the right torque.