Grammatical Errors

All of them can be correct.  In this context, sooner means exactly the same thing as rather.  Perhaps it is considered slang, but I would say that in the US it is more common to say that one would sooner die than to do something the person would hate to do.  Unless one is being held as a prisoner of war, the speaker doesn't actually expect to die.  It is a way to express how very much they don't want to do the other thing.  The last example you gave would be used in a slightly different scenario.  If the speaker will get in big trouble with someone else if they confess it, they might say they better die - or else be killed for it.  Again, this is meant figuratively.