Isn't an article "a" or "the" necessary before "trouble"?

What does "have" mean here? (have got to, has got to)

"Have" here comes from the phrase "have got to." This expression is using the present perfect tense (have got), but you will find that this is a common expression that has the same meaning as "have to" or "must." Ex: I've got to make a decision. Ex: Did you have got to do?

I thought "come from" is used when people talk about their nationality. What's it mean here and is it used in other contexts too?

Good question! "To come from somewhere/someone/something" means "to be from ______(place, person or thing)". It is used to find out the origin or source of something (for example, someone's nationality is their origin). Max wants to know the source/origin of the tickets. In this context, the source is not a place but a person (Moon). Ex: "These tickets come from Mary!" Ex: "Mary comes from Sydney, Australia."

Does the phrase "worse off" emphasize how desperate the situation is? Can't I use "worse than" instead?

The phrase "worse off" means to be in a worse situation than something else. "Worse than" expresses a similar meaning, but you are correct in that "worse off" emphasizes the desperation of a situation. So you could use "worse than" here, however, "worse off" makes the situation sound more dramatic and indicates that Europe had it way worse than something else. Ex: As a single mother of five children, she's worse off than you. Ex: Even though we have no money to give, we should help them because they are worse off.

What's "sorority" mean? Does this generally imply something when used in this kind of context?

A "sorority" is a typical North American female society in college or university. But you're right. It does imply something. There's a kind of negative stereotype about "sorority" girls: they love to party and drink, they like material objects, they're naive, and they only really care about themselves and the other people in the sorority. They're supposedly very cliquey. So I wouldn't say it's a positive thing that he's calling her a "sorority girl." Ex: I would never have thought that you were in a sorority. You don't seem like the cheerleader type. Ex: Do you want to join a sorority next year?

What's the main difference between saying "greeting" and "greetings?"

Good question! The plural form "greetings" is used as a word of welcome or hello. You wouldn't use the singular "greeting" to greet someone. "Greeting" in the singular form can refer to a hello, or it could be the present participle or gerund form of the verb "greet." Ex: Greetings, friends! How are you all? Ex: I am going to be greeting so many people today at the event. Ex: What a lovely greeting she gave us.