Is "put a sock in it" used commonly in a daily conversation?

Which one does sound natural? "think happy thoughts" vs "think happy things"?

"Think happy thoughts" is a very common and well-known expression, therefore it sounds more natural than "think happy things".

Shouldn't it be "there're" since there's the plural form "costs" at the end of the sentence?

Yes, you're absolutely correct! "Costs" is plural, so "are" should be used instead of "is" here. You've actually pointed out one of the most common grammar mistakes made by native English speakers, which is to use "there's" to refer to plural nouns. Ex: There're so many cars on the road today. Ex: There's a train arriving in ten minutes.

Can I think "just me" same as "alone"?

Yes. You absolutely can think of "just me" in the same way as "alone" in the context of this clip. Monica is trying to convince Rachel she was eating lunch by herself. The phrase "just me" is also sometimes used as an expression to mean that something is my opinion; my perspective; my thoughts. Ex: I’m not into raw oysters but that’s just me. (My opinion) Ex: It was just me at home. (I was alone at home) Ex: Is it just me or is it really cold in here? (Is it just my opinion) Ex: It's just me. (I am alone)

Can I use "cutie" to boy as well?

Yes, you can. "Cutie" just means that you find someone attractive or endearing, therefore it is gender neutral and can be used for men and women. Ex: My boyfriend is such a cutie. Ex: I love being a kindergarten teacher. All the kids are such cuties.

Would it sound awkward if I say "studied" instead of "practiced" in this context?

It might sound awkward, yes! That's because he's already studied the language, and now it's just a matter of practicing speaking it. If he was still learning the language, then "studied" would work well. Ex: I haven't studied French in a while, so I can't remember a few grammar points and vocabulary. Ex: I haven't practiced French in a while. So It might sound a bit awkward.