It's not the same as "I got the bus home", if that's what you're thinking. :) "I got the ticket (to get me) home."

Yes, it is correct. the ticket home = 回家的票

I suppose, its correct in the right context. For example if you were catching a bus, train or a plane home then you would say "i got the ticket home" to a friend as its less formal. But if you want it to be more formal " i got a train ticket to go home" or "i have the train ticket to get home"

Get a dictionary and learn something about "part of speech".

需要根据上下文判断 直译 我有回家的票了。 意译 可能是具体的票无关， 此处更像可能性， 机会。 所以可以理解为 我可以回家了。