Is “funnily enough” an idiom? What does it mean?

What does "break" mean here?

Here, "break" means "convince". The writer of the note (Monica) is saying she knew Phoebe and Rachel would convince Chandler to look for presents with them, even though he knows he's not supposed to. It is implying he is easily convinced.

What does "crazy nasty" mean? Is "crazy" used a lot to emphasize the adjective?

You are correct. "Crazy" here is being used to emphasize the adjective "nasty". Informally in English the word "crazy" can be used to mean very, really, extremely or intensely, so it acts as an intensifier. Ex: Well, that was crazy stupid. Ex: She is crazy beautiful.

What does she mean by saying Monica with “your steady hand”?

In this video, when Rachel says to Monica "your steady hand" she may be referring to how calm Monica is about switching apartments with the boys. Rachel is having a hard time staying calm in this situation because she doesn't want to switch apartments. Monica is somehow staying calm and collected even though she has to move into the guys' apartment.

What does "stable" mean in this context? I think it does not have the same meaning as the verb "stable!"

A "stable" is a building in which animals, especially horses, are kept. It most commonly means a building that is divided into separate stalls for individual animals. It does not relate to the adjective "stable" which means that something is consistent and unlikely to change its state. Ex: We bring the horses out of the stable three times a day so they can walk around. Ex: I visit the horse stable on weekends so I can see my horse and ride him.

Is “funnily enough” an idiom? What does it mean?

"Funnily enough" is not an idiom. It means the same as "surprisingly". It is used when something surprising is true. Ex: Funnily enough, I am the only person in my family who doesn't like watching TV. Ex: Funnily enough, the person I'm dating is completely different than me.