What does "paint on [something]" mean? Is it a metaphor? If so, in what circumstances could I use it?

What does "another round" mean? When can I use this expression?

"Another round" is used to at a bar (a place that serves alcohol) to order more alcohol for you or you and the group you're with. Here, however, he is saying "another round" at a cafe to order more coffee to be humorous. You would not order more coffee this way, this phrase is traditionally only used to order more alcohol.

What does it mean "hard-core" in this case?

To be "hardcore" means to be extremely dedicated to, committed to, or intense about something. In this case, she is saying that he is displaying a "tough" persona without any vulnerabilities. Ex. He's so hardcore about fitness, I've never seen him eat a pizza or burger.

What does "hit sweets" mean?

Here, "hit the sweets pretty hard" basically means that the boy eats a lot of candy. "The sweets" means sweet things, like sugary drinks, candy, and chocolate. Ex: My favorite thing to do is to hit the sweets while watching TV. Ex: I hit the sweets less than I used to, because I got too many cavities.

What does "paint on [something]" mean? Is it a metaphor? If so, in what circumstances could I use it?

This isn't a common phrase, actually! I think it was made for the song specifically. Generally, to "paint on [something]" means to literally paint on something. It could reference covering or hiding something with paint, too. Here, it could be a metaphor to put or paint "caution into the wind." "To throw caution to the wind" means to stop being careful and do something slightly dangerous. "Paint" just sounds a lot more poetic and gentle than "to throw." Ex: I prefer to paint on canvas rather than paper. Ex: After thinking about starting her own business, she finally threw caution to the wind and did it.

Is "Pheebs" a nickname for "Phoebe"? How do you make the nickname after the name?

Yes, "Pheebs" is a nickname for "Phoebe". It is not a common nickname, but just one that they have given to her personally. You can make any nickname from a name by shortening the sound of the name or adding something else to the name. Ex: My name is Elijah but most people nickname me Eli. Ex: Her name is Madeline but she often goes by "Mads".