"For how long are you going on your trip" is completely correct, but it definitely sounds formal. You could use it with no problems. The other two options are also completely acceptable and are probably used much more frequently. You'll likely hear at some point: "How long are you going on a trip for?" This is incorrect, as a preposition ("for") should never go at the end of a sentence, but many people don't care and do it anyway. Still, the two more natural examples you gave are probably more likely to be used.

You can ask all three. The first is more for when a trip is being planned or thought about. The other two would be when a trip has been decided on, and you want to know the details.