What does “be in one’s feelings” mean?

I learned to put "the" in front of superlative like "biggest." But when I use a possessive form like "your" in front of it, is it okay to skip "the"? (Ex: My smallest dream, his fastest car)

Yes, it is okay to skip the definite article "the" in front of a superlative when there is a possessive pronoun and you are relating the superlative to someone else. Ex: His greatest strength is his kindness for others. Ex: My happiest memory is my wedding.

"Well" is not used in a literal way here, right? What's it mean?

"Well" doesn't really have a meaning in this situation. It's often used as an exclamation to continue a conversation or change the topic, or when someone is pausing to consider what they will say next. In this video, it looks like Joey has used "well" here to introduce a new topic of conversation. Ex: Well, that was the most boring movie I've ever watched. Ex: Well, don't ask me. I don't really know either.

I assume "migrate" and "immigrate" are two opposite words. Then what does the prefix "i-" mean?

Actually! They're not quite opposites. But the words are related! "Migrate" means to move from one place to another. Whereas "immigrate" means to move from one place to another to live there permanently. The prefix "I-" or "Im-" in this case actually means to "enter into." The opposite of "immigrate" is "emigrate," which means to leave a place, as opposed to entering a place. Although, they're used interchangeably. Ex: Whales migrate toward colder poles during summer. Ex: The family is planning to immigrate to New Zealand next year. = The family is planning to emigrate to New Zealand next year.

Why would Morty mention "first part" in this context?

He mentions "first part" because he's hoping that Rick doesn't stay a pickle, as if a "second part" of a magic trick would turn Rick back into a human! Ex: For the first part of my magic trick, I will disappear. If the second part works out, I'll reappear soon. Ex: Was that the whole story? That was a bad ending. You need a second part.

What does he mean by "I'm down" and "I'm out"?

By "I'm down", he's referring to being emotionally "down"- depressive or sad, not feeling like himself. "I'm out" has a few possible meanings that could all work here! The first is similar to "I'm down", but it also means "out of the picture"- he's no longer part of the other person's life. "I'm out" could also have a figurative meaning of being "knocked out", as he can no longer "stand" on his own as if knocked down in a physical fight. Ex: I've been feeling so down lately. I'm not sure what to do about it. Ex: I can't be in your life anymore, Ashley! I'm sorry. I'm out. Ex: I'm tired of arguing, John, I'm out.