I have been training for the CRO PFT, and I want to know if flip turns are permissible and which method most people use/used. Let me know!

Not sure if flips turns are permissible, to the best of my knowledge, everyone did open turns at phase 2, we had to wear our fins too, so it would make a turn more difficult... aka grab wall (head above water), push off with legs and feet.

Here's a video of a PJ swim. Note how he turns. That's the turn everyone does. This is a "Fin swim". During the PFT, you can use your Arms.

http://www.youtube.com/watch?v=V9w-L85511w

can you use dolphin kick (instead of flutter) the whole time instead of just 2 times of the walls, for "fin swims"?

no. 2 dolphin and that's it. 1 leading arm.

they want you to do "wide powerful strokes" (with your legs) at Indoc.

So what type of swim is the absolute fastest for the PAST? The style used in that video or freestyle or what?

Typically it's the freestyle but the obvious answer is: the style that is fastest for you. Do 500m using the allowed strokes and you'll likely have your answer.

If you're doing the PAST or CRO PFT, my personal recommendation would be to fin as hard as you can and use your arms (you're allowed to wear fins on the CRO PFT), but on the PAST, it's a finless swim, so freestyle would be the best. Make your turns are really quick too, touch and go as fast as you can, you can really waste a lot of time pulling yourself up and pausing on the side of the wall if this is a temptation for breaks/breathing.

If you're doing the official "Fin swim" at Phase 2 or indoc, of course, you cannot use your arms. You'll just have 1 leading arm to point in the direction you are going.... that swim can be up to 8000m