I have been playing this game for a couple of days now and i really dont understand the story with Basch, its confusing and nothing like the other games.

My question qould be, how did he "betray" the king or how did he get set up for it? Why did they let him live?

he didn´t kill the king, not even Vaan´s older brother!!!
he is just a guy that was in the wrong place in the wrong time.

So his twin set him up ?

Why was he kep alive in that dungon?

His twin brother set it all up. as for the dungon he was kept there also because of his brother.

Please mark spoilers, everyone.