Running and then dies

Hey guys, this might be a very stupid question but here it goes. So i decided to take my car out for a spin, it gives me a little trouble starting up but all is good. I move about 20 feet and the engine just stops. no bogging just straight up dead. So i put it into neutral crank it, it starts.
Go about 1/2 a mile and i'm at a stop sign and again, dies.
i crank it, starts then dies, try it again same result. third time i turn the headlights off and walllah it starts but i gas it a little to keep rpms up. all is good.
I pull over and check my gauges and i notice that my temperature gauge is and C all the way, so i let it idle for a while, and then its fine.
I didn't have far to go but, could Low temperature make the car die like that?
I live in southern california, it was 60 F out.
Also, my belt for my alternator is slightly larger than the pulley itself, ever so slightly, and sometimes when it starts it squeals if i gas it at the start, but as soon as i take off my foot from the gas its ok. Could that also be causing problems?

Squealing fan belt is not good for an alternator but you would think that it would cause a heating problem.

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No i just thought that maybe it was a sign of not enough juice being created by the alternator.
Is there any way to get a pulley off without an impact gun, I don't own one; and I have the pulley from my old alternator which should fit snugly.

Actually a while back (before i got a new alternator, and before i had tuned the carb a little i had to gas it in order to start up) but now usually it starts up without a gas, besides the initial gas pump)
It was the first time this happened to me actually, where it'd start then die then start then die then start. Usually on a bad day (once every 5-6 months) it'd die, and start right up. I think it was just a lack of a warmup in this case. I drove it back home after and it ran well.


Also, my headlights dim a little at lower RPMs, at first i thought it had to do with the stereo system i had in the car, and it probably does, but what would remedy this? a relay or a capacitor? the alternator is 130 amps, and the amp is 600 W; running 2 components; 2 6x9s and a sub