[Tzeentch Lord]

When a unit that is immune to psychology breaks from combat, do they automatically rally the turn after, order do they still have to make a leadership test to rally despite being immune to psychology (and hence immune to panic)?

Psychology and Break Tests are two completely different things, being immune to psychology or panic doesn't exempt you from taking break tests. I don't believe that troops who are immune to psychology automatically rally, I'm pretty sure they have to pass a leadership test to rally just like everyone else.
Fleeing because of panic (or fear) and fleeing because of a failed break test are just two different ways of causing the same effect (ie. fleeing), it's just that troops who are immune to psychology or fear can only end up fleeing by failing a break test.
I hope my ramblings make sense!

I know that skirmisher don't negate ranks when charging a unit in the flank or rear, but if they number US5+ do they still get the +1 or +2 bonus for a flank or rear charge?

Yes, they do.

can fast cavalry march within 8" of the enemy?

Can't remember off the top of my head, but have a look in the rulebook when you get it back and check the pages about Fast Cavalry, if there isn't anything about being able to march within 8" of the enemy, then they can't do it!