All the guages, etc. are grounded under the dash. If the headlights don't dim when the guages go out then the underdash grounding points are bad.
The car not starting could be a whole other issue. That can't be the alternator because if it was the symptom would be a dead battery and you're battery's not dead because it starts after a few minutes. That could be from a bad starter solenoid, a bad ignition switch, or a bad ground among other things.
The car does not require a battery while running. It needs either a battery or a functioning alternator. A car can run off of only one or the other but if the alternator's bad you won't get far on a battery alone.
When the electrical system messes up while driving pay attention and try to list as many things as you can that shut down or act odd. Does the engine run fine, if it's an automatic does it shift fine, etc.
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