Music:
Snowy. I found a playlist for the Undertale OST! Isn't that wonderful? (Yes, it is, the music in this game is fan-damn-tastic.) As always, open the video in a new tab while you read; shift-click or wheel-click is a Chrome shortcut that may work in other browsers, idk.

To save some explanation: it's an inventory item storage. As I find out rather later, you can actually call the box on your cellphone to access the extended inventory space it provides. Aside from limiting the number of items you can carry into battle, it seems like a weird way limit-sort-of-but-not-really the stuff you can carry.

But the first time you open it, you discover that someone else left a weapon upgrade inside. The Tough Glove is a subtle pun and a slight boost above the Stick you start the game with.

One screen up is a fishing line. Reeling it in shows that someone, uh, is trying to find a date in one of the local sea creatures. His or her phone number is attached, but not added to the list of people you can dial on the cellphone.
Probably just a catfish account, anyway.

So, the local wildlife. Each time the Snowdrake attacks, it belts out a pun that makes that "catfish account" line seem sophisticated. "Ice to meet you!" or "My fave cereals are 'frosted'" or "Ice puns are snow problem!" To appease it, laugh at its jokes. ("See? Laughs! Dad was wrong!")
It gets pretty annoyed if you start laughing at it before it tells a joke, but then realizes that its own name is a pun and has a minor existential crisis.

Ice Cap is utterly fascinated with its own hat. Everything it says and does centers on its fantastic headwear. If you flatter it, it'll lower its guard, allowing you to
steal its hat...

...and an Ice Cap without the Cap part is just Ice. If you tell him that he still looks okay even without the hat, you can let it go. (If you try to steal the hat without flattering Ice Cap first, he'll call for help from the Fashion Police.)