What's "enter [someone]" mean? When's it used?

Is "last but not least" only used in speech?

Not at all! Although it's more commonly used in speeches to give thanks to someone, or to introduce a list of items, you can use it in writing as well! Ex: Last but not least, I'd like to thank my parents for their love and support. (Speech) Ex: Last but not least, the weight of the item is crucial to calculating velocity. (Writing)

Can I say "of" instead of "about" in this context?

Although "of" and "about" have the same function, when you are asking someone about their thoughts, "what are you thinking about" sounds most natural! Ex: You look lost. What are you thinking about? Ex: I'm thinking about what I should do after work.

How come two nouns "others" and "something of value" come in a row?

The nouns "others" and "something of value" come in a row because he is speaking in a shortened way because he is reciting a list. His full intended phrase is "You should serve something of value to others".

What's "enter [someone]" mean? When's it used?

"Enter [someone]" comes from stage directions to indicate when an actor is coming on to the stage. Now it's used to indicate when someone is part of a conversation or story as if to say, "This is where [someone] comes in," indicating that this is a key part for them. So it does have a slightly dramatic effect when it's said, but it's fun to use. You can use it to indicate surprise, coincidence, or opportune time. Ex: I had lost my puppy. Enter Dave, who offered to drive around and find the puppy. Ex: The party was getting so dull, then enter Jen: she lightened up the whole place.

How is it different between saying "get to shake" and just "shake"?

To "get to do" something means to have the opportunity to do something. When we use "get to" it is to emphasize the face that we want to do this thing but there was no chance to. In this case, Chandler wants to "shake his belly" but he won't have the chance to. Simply saying "shake" here would miss the nuance. "I didn't shake my belly" doesn't give any information about why he didn't. Whereas "I didn't get to" lets us know that it wasn't his choice to not do it. Ex: My whole family gets to go to Disneyland this summer. Ex: I didn't get to eat anything all day. Work was so busy.