I don't understand what he means by "You ain't gotta be my lover for you to call me baby". Does it mean, "You don't have to be my lover for you to call me baby"? If so basically is he saying, you can call me baby?

Is this sentence's grammar correct? Can I say "Not only he has homered" instead?

The grammar is correct in this sentence, "not only" can be used at the beginning of a sentence to convey that "there was not only one thing, but there was an additional thing". Another example is: Not only did I make dinner, but I also made dessert. You can also say this sentence as "He has not only homered, but he's (also) tied the game."

I don't think 'what is it' here literally ask what something is. I wanna know the meaning and when it can be used.

You're right. "What is it?" is not being used literally here. English speakers can use "What is it?" to express: "What's wrong?" (showing concern) or "What do you want?" (showing slight annoyance). How it sounds depends on the tone of voice you use. Ex: What is it, honey? You look sad. Do you have something you want to tell me? Ex: A: Boss, can I talk to you for a second? B: What is it? I'm really busy right now.

What does it mean to do something "actively?"

"Actively" means deliberately, 힘, positively. Here, "no one ... actively hates me" means "no one vigorously/especially hates me." However, in many contexts, "actively" is used to describe action - it means to take the initiative to do something. For example, if a company is actively trying to recruit you, it means they are taking the initiative to get you to join their company. Ex: I don't actively look for new friends, I just seem to meet new people naturally. Ex: Are you going to actively pursue this man? He seems like he's hard to approach.

Does "bit" necessary in this sentence? What does it mean?

Yes, "bit" is necessary here. "Bit" here means "part" or "section". Ex: I liked the bit in my book that talked about cooking. Ex: Please act out a bit from a movie for your audition.

What does "take down" mean? If it's not a phrasal verb or something, then why is "down" used here?

Yeah, it's not quite a phrasal verb here. Here, "take" is similar to "bring," and "down" is an adverb that indicates a place that is some distance away or towards the south. Usually, to "take [someone] down [somewhere]" means to bring them to that place. Ex: I'm going down to the shops quickly. Need anything? Ex: She's down by the harbor sailing her boat. Ex: I'll take her down to the school for volleyball practice.