1975 Dart Sport - Alternator Issue?

Looks like the original alternator. Way cool! cool-gif.gif
There might be assembly date and part number stamped in the rear casting, and the tag on the front half is the factory identification info. Its color coded according to amp 'rating'.
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The previous owner installed an electric fuel pump, new starter, gps, and backup camera(I have since disconnected the gps).
This is almost certainly the cause of all the problems. (a) The electrical system was not really designed for extra electrical loads at idle. (b) the alternator was not designed to produce the additional power needed for these accessories. (c) The method and location of tapping into the existing system to add these things may be be contributing to issues.

The car was missing the dome light bulb. About two weeks ago I installed the dome light bulb. The bulb comes on when the doors open. However, after installing the bulb the car would hardly start and once it started it ran really rough. I removed the bulb and the car was able to start and after a minute or so it ran fine. My initial thought was that the bulb had a short or something and was pulling alot of current(not an electrician).
A bulb can have an 'open'. It is possible to have a short in the dome light wiring. It would have to be at or before the dome light on the feed side. If so, the fuse should blow.

I tested the battery with my multimeter and it checked out(12.5 volts or so)
A little low. Not terrible. 12.8 V would be better.
I hooked up my multimeter and the battery was around 14 volts. With the car running I put in the bulb and turned all the accessories on(ac, lights, left doors open so bulb is on) and I hooked up my multimeter. The battery voltage slowly dropped until it got to around 12.3 volts and the car started to run rough. I took the bulb back out and again the car ran fine.

Today I ran the car with bulb out and turned all the accessories on. The battery voltage was initially around 14 volts but as I let it run it got down to under 13 volts.
You've just proven that the alternator is not producing enough power to run all of the accessories at the same time. Or possibley, a poor connection in the output wiring is getting hot, and the hotter it gets, the more resistance it has to flow.
In either case, the problem is not the dome light per se.
The one electrical load we don't know in either of your tests was the battery.
That can be a big variable. If its really low on charge, it could be sucking 20 or 30 amps at 14 Volts. On the other hand, if it was fully recharged, it would draw nothing measurable at 14.0 Volts.
The ammeter will indicate whether the battery is charging or discharging assuming the electric pump and camara was wired into the fusebox.
If that stuff was wired to the battery positive or the junction at the starter relay, then that's not true any longer.

One site I read told me to test for ac voltage across the terminals to determine if there is a bad diode. I set my multimeter to 200 ac volts and I got around 30 volts ac across the battery terminals.
Well that can't be. So maybe just a bad test.

My suggestion is:
Figure out where the electric pump and back up camara are wired in.
Make sure the dome light is still properly fused.
If it was me, I'd put a mechanical pump on it and also remove the back up camara. These cares have pertty good viewing all around. But the backup camara is really not a problem itself. It should be wired into the accessory side of the fuse box.

As best I can tell, '75 wiring is still somewhat like this. (76 is very different).
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Important things to notice are:
The ammeter is located to show current flowing in or out of the battery.
The fuse box has some circuits directly wired to the battery and alternator. One of these is the dome light.
The box also has circuits that are only turned on when the key is in Run or Accessory.