Are the rules different based on league? Youth > HS > college/pro?

I read on a site or two that all fake throws to 3rd are illegal, now, but couldn't tell if that was across all ages levels or not.

Faking to third is not allowed in MLB or NCAA. I know it is allowed in Perfect Game and I think still allowed in NFHS. So yes, different leagues have different rules for this.

That being said, I don't know how a left hander can fake to third illegally. He would have to step off to legally go to third, which at that point he's considered a fielder and and is not required to throw it.

Is a jump-move, similar to a RHP picking to first, considered stepping off?

Depends what your definition of faking a throw is. A LHP would have to step off the rubber in order to pickoff to third which in turn allows the pitcher to do whatever the hell he wants after stepping off

Does a RHP have to step off the rubber to pick off to first?

Yes, sorta. Again a grey rule where they "step off" but actually don't.

The back foot needs to come up and step behind the rubber before the front foot leaves the ground to be a legal disengagement of the rubber and not require a throw (to first or third).

If the feet are moving at the same time in the air then you haven't properly disengaged the rubber and therefore must throw the ball.

The same is true for the little jab step you see pitchers do with their back foot where they just pick it up and rotate it slightly but don't move it behind the rubber. That also isn't a proper disengagement and therefore also requires a throw.