Situation 1

It sounds like F1 is standing in an open stance meaning his foot is slightly towards firt so he can see the runner without turning his shoulder. This is fine. From there he can come set either open or closed (his lead foot towards the plate). Once he closes his stance he can't turn his shoulder open again. If he does it's a balk.

Situation 2

The tap towards third and back to first seems like a balk. From your description it doesn't sound like he broke the rubber. I don't have a FED book handy but in OBR 8.05 note it says he can't,while in contact with the rubber, step toward third and in the same motion turn back to first and throw there. If he makes the same type move but breaks the rubber then he's fine.

Situation 3

If he turned toward second and then ran toward the runner he had to land and that would have to be in the direction of second. He doesn't have to make a fake at all. This sounds like nothing to me.

Situation 4

See three. Here again in the turn he has to put his foot down somewhere and it sure sounds like toward second to me. He doesn't have to make the fake.

Situation 5

This question is a little ambiguous. If you have R1 and he breaks toward second the pitcher may turn and throw to second to make a play. I'm not sure what you mean by stepping toward the runner is.

I hope I've understood what you were asking and if not try another explaination.