1) It's a phrasal verb, based on the verb "to go:" "to go live." However, it's common in English to blur the boundaries, use verbs as nouns or nouns as verb. In this case, by joining the words with a hyphen, we show that we are using it as... I would say an adjective here. Your usage "go-live date" is fine. It is somewhat informal or colloquial. I think a more formal phrase would be "release date."
2) When you are using it as a verb, you can use it the same way as you use the verb "to go." "My app went live" is correct, and so is "my app has gone live."
3) Yes, they are the same. "Live" is actually an old term from radio broadcasting. A "live mike" is one that is "on the air" and broadcasting now. A "live broadcast" is one that is happening now, as opposed to a recording. By analogy, your app is "live" when it is on the Internet and everybody can see it.