What does "cozy" mean?

Isn't "caveat" a synonym for "warning?" Why would she say that?

Yes, that's right! Here it's she's kind of like she's "warning" the audience about a detail that needs to be considered when evaluating or doing something. It's an extra detail she and the viewers didn't consider fully before now. Ex: I'll leave you with this caveat: Never go ice skating on thin ice. Ex: She agreed to do the interview with the caveat of asking her own choice of questions.

I've heard "so like" and "you know" so many times when talking with natives speakers, but those don't really have any meaning. Is it like filler words?

Yes, you're right! Native speakers say things like "so yeah," "so like," "like," "you know," "you know what I'm saying," and many other similar expressions commonly. They don't have any particular meaning, they're just filler words! Ex: It's like, so cold outside today. Ex: So yeah, I don't feel that great today. Ex: It's not even that late, you know, stay a little longer!

There is no be verb, but I don't know how it becomes future tense.

It should be "You are going to fit in here" but since this is a skit, they changed it to "you gon fit in just fine here." They did in fact skip "are" in this sentence. Even though this is grammatically incorrect, you do hear this type of slang used occasionally in spoken English.

Why use "get hurt"? Why not use "take hurt"?

To 'get hurt' is a 'phrasal verb'. 'Phrasal verbs' are "phrases that indicate actions". They are generally used in spoken English and informal texts. So to say 'get hurt' instead of 'take hurt' is a grammatical rule in English. Also 'Take hurt' sounds very unnatural in English. Ex: "Jessie didn't want to go skiing with us because she was afraid that she might get hurt."

What's "breath mint"?

A "breath mint" is a candy that is mint flavored and is meant to be sucked on to "freshen" your breath and make it minty. It is often used after eating, especially if you ate something with garlic or that's spicy.