How long will the iPhone 5 Video camera run?

Does the camera stop after a certain amount of time?

IT should run as long as it wants. Depends on memory and how long battery life.

Will the camera stop before the phone runs out of space to store the video or will it just run continuously.

It will stop before it runs out of memory.

When stopped later how will it handle a video that may be too large to store?

The video will be stored in your Camera Roll

If the phone does have a set ammount of time limiting the length of a video, is there a way to override the timer so that the phone will run continuosly until stopped by me?

Unless its jailbroken.

If you do this, be aware the video files will be very, very large. You may have difficulty moving them to your PC. Do some trial runs first.