These are probably obvious questions, but our school just got VEX running a few days ago.

Is it possible to run the robots without RobotC running? Like, does the debugger have to be open, or can the robot be fully autonomous?

Can two robots be programmed with a single seat of RobotC?

How is VEXNET used? Does it go through the remote or something?

Thanks in advance.

No, ROBOTC does not have to be running. If this were the case, then no competitions could be run. Turning the robot on will start your program (If you don't have a joystick attached, then it will take a couple of seconds while the cortex searches for the joystick).
You can program as many robots as you want with one license.
VEXNet is used through the little black keys that say VEXNet on them. One is plugged into the joystick, and the other is plugged into the cortex. You will find information on how to connect them in the packaging and online on the vex wiki.

Since you're a newbie, I'd recommend checking out the VEX Cortex Video Trainer. Lots of free tutorials on setting up, using motors, sensors, remote control, ect: http://www.education.rec.ri.cmu.edu/pro ... index.html