If I push the limit with Vader (the TIE Advanced ship card), will he perform 3 actions? Thanks!

Assuming you trigger push the limit off of his second action, yes. If you use PtL off the first action then you will gain a stress from PtL, which will prevent the third action (from Vader's ability) from being possible.

Keep in mind that each action has to be different, and that the PtL action must come from the action bar, not an upgrade or damage card.

Also remember that some upgrade cards actually add the action to your bar (e.g. Engine Upgrade) and can be used through Push the Limit.

This is a separate but related issue: if you use advanced sensors, you cannot trigger Vader's pilot ability, but Push the Limit would be the easiest way to get a second action.

Thanks!