You've got the point exactly, Vitor. If there aren't 'any verbs around' we use the object pronouns - at least in modern-day English. Compare: So do I. Me too. Neither have I. Me neither.

Sure, I would say "Us and them" although, for all I know, that song may be why people use that particular construction. It was a very popular album for a very long time after all. "Me too" is also very common.

Yes, they're correct and standard. Surprisingly, as native speakers, we don't even bother questioning this. If we really want to think about it, I'd say there's an implied preposition or fuller phrase somewhere, or some context which requires the object pronoun. (This song is about) Us and Them. (The matter is between) Us and Them. (The same goes for) Me, too. ...or even... "I like bananas. What about ____?" We would never say "what about he/she/they/I etc". The response relies on being an elliptical sentence, ie. not a sentence in its own right, but a fragmented response. Using a subject pronoun makes no sense here, unless you answer with a complete sentence, as in Evelyn's example. There's also the possibility that we've borrowed a bit of grammar directly from French. English does that. The response "moi aussi" is correct for agreeing to a question like in your example. It makes no sense at all to reply "j'aussi".

It's perfectly fine to say, "Me, too." You could also say, "I like them, too." That might make you feel more comfortable in your conversation.