What does 's stand for?

How to use the expression “all about something”?

If you are "all about" something, it means that you really like or enjoy it. Ex: I'm all about basketball, it's my favorite sport. Ex: She's all about reading, I haven't ever seen her without a book.

Why's he talking about internet here?

Although I can't be certain, I think "go home ... high-speed internet" is talking about the actions he takes to isolate himself from others, so that he can be alone in his sadness and loneliness. It sounds as though he'd prefer to go home to be alone and go online than to be outside with others. It does feel a bit random to mention "high-speed internet" suddenly, but if you consider the rest of the lyrics, it probably is referring to his isolation from the world.

Isn't it double negative? What does it mean?

Yes it is! "Didn't never" is definitely a double negative and is incorrect grammar. It means "didn't ever." The sentence should be "For some reason, Jenny didn't ever want to go home." If you have ever watched the movie Forrest Gump, you may notice that the character Forrest speaks incorrectly throughout the movie. This is done to fit the character Forrest.

The expression "come on" seems to have different meanings or nuances depending on the context. Could you explain?

You're right, "come on" does have a few different nuances depending on the context. In this context, "come on" means the same thing as "let's go" or "hurry up". You can tell it means this here because they are rushing around and trying to avoid danger. In other contexts, "come on" may be used to express anger or annoyance towards someone else. In those cases, you can tell that someone is using it to express annoyance based on their tone of voice and the stress of the syllable "come" instead of "on". Ex: Come on! We're going to be late for the meeting. Ex: Oh come on, you seriously didn't hear what I just said?

How come "desperately" describes "neened"? not "said"?

‘Desperately’ is an adverb meaning ‘in a way to show despair.’ or ‘to seriously degree’. So, in this case, he’s saying that the math club seriously needs to members. Ex: I desperately need a job. Ex: He desperately needs a drink.