Hi, great questions!
1) Yes you're right - the 'big quad thing', a.k.a rocket carrier a.k.a gondola, stabilizes the heading of the rocket.
2) It does not necessarily imply that, but yes that would be a great way to do it so that the rocket control system doesn't need to correct for the tumble.
3) There are multiple ways to pitch up the rocket, and you are right that one of them is to use the props. Alternatives are to have thrusters on the rocket or have gimballing on the rocket nozzle. All are viable approaches.
4) This is a complex question as we have many variables to play with, such as choice of launch altitude, size and design of propellers, etc.