What does "Bunk over" mean?

Here is a quote from The Avengers, 2012:

• Stark : The next building is gonna say "Potts" on the tower.
• Pepper : On the lease.
• Stark : Call your mom. Can you bunk over?

Q. "The building is gonna say "Potts" on the tower" What does exactly it mean? Maybe it means "The building is gonna be hung out a sign written her name, Potts on the tower?"

Q. Pepper said "On the lease". She meant "You should sign on the lease contract"?

Q. Lastly, I have no idea what Stark's last line means, specially "bunk over" And why Stark wants Pepper to call her mother suddenly. Do you have any idea on this?

Stark : The next building is gonna say "Potts" on the tower.

"This building says "Stark" on the tower. The next one is going to say "Potts" on the tower"

They're talking about the "next building" after Stark Tower, aka the Tower that has Stark written in big letters outside. So yes, he's saying let's have the next building (with arc reactor energy etc) be Potts Tower, with Potts written in big letters on the actual building.

Pepper : On the lease.

"The next building is going to say "Potts", but only on the lease"

Pepper, otoh, has no interest in having her name in big letters outside a building. She's saying the next tower will indeed have her name on it, but only on paper - she'll own it, but she feels no need to see her name splashed on the outside. Edit: It might also refer to Potts not wanting the next building to be named Potts Tower either, for much the same reasons.

Stark : Call your mom. Can you bunk over?

"Call your mom. Can you stay over?"

This one is purely tongue-in-cheek. Stark is basically just asking if she can stay over / not go to her own place / business meetings / whatever, but doing so while acting like they're teens and she needs her mom's permission in order to stay over.

It means:

To stay the night; sleep: bunk over at a friend's house.