On PC, can you play with a LAN? If so, can you go against your friends, instead of helping each other out?

I'm pretty sure it's been confirmed, but even if it hasn't, it wouldn't be hard to include in a multiplayer game, I would be surprised if it wasn't there on release.

There are many games that do not support LAN, but that support gaming over the Internet. So the risk exists. But I have not read that there should be or not be.

I tried to find whether there is a "system link" for xbox 360 but found nothing. If there is system link, there is almost certainly also LAN. Check if you can see anything about it.

It happens, but generally not when the game is made by a developer like SD that really enjoys supporting the competitive community. Which is why I expect it to be there.