Check the compression on each cylinder. Also check for proper valve operation. You may have a flat cam lobe or bad rocker arm on either the intake or exhaust side which will give the rough idle you are experiencing.
Check the fuel pump relay first. It could be sticking. Under the hood fuse box. Little square switch module. Usually labeled on the lid of the fuse box. To test it,
Swap it with the horn relay if it's the same kind. (Usually is on gm/chevy) start it a few time to see if that changes anything. If not then it could be fuel pump itself
old carb type do suffer from flowt problems, also people dont change the filters regularly so the inline carb filter blocks up., you would need to remove the carb and take the float chamber housing off. check the condition of the floats, and check the valve, if there is what looks like wet/dry dust in the float chamber then the filter has failed and you will need to clean the jets, you will need a few cans of carb cleaner or a big can of spray to clean it out. change the filters depending on carb fitted, there can be a filter above the valve or the fuel in line to the carb, also the main filter. replace the seals and refit when clean, would also at some point remove the tank and clean it out.
The belt should not come off. If it didn't come off before the new alternator was fitted it should not came off now and the fact it does should not be your worry but the worry of the person who specified the repair, supplied the parts and carried out the repair. You presumably paid somebody to cure a problem and not to provide you with a whole new problem.
Clearly the new tensioer has not cured the problem...
If the pulleys are undamaged, perfectly round and the alignment is correct and the belt is tight enough, it should not come off.
If the pulleys are ok and the belt is coming off then the belt isn't tight enough. I assume the tensioner is a spring loaded automatic type?
If it is a spring loaded type, it indicates something sometimes is needing so much power to keep it turning it is overcoming the tensioner spring pressure and creating a slack belt.
Sorry but other than that mad idea I have nothing to offer...
Sounds to me you have high resistance problem , Main power feed to the alternator . Check the power junction block behind the brake booster on the fire wall . There is nothing in the instrument cluster to regulate voltage .B+ voltage to battery symbol when key is on , then when it starts voltage from voltage regulator inside alternator goes back on that circuit shutting off the light . Voltage on both sides of the light cancel's each other out . Yes, i would look for a burnt connector . at the junction block . Do a voltage drop test while the vehicle is running . Voltage Drop Testing the Charging System
The flasher for the signals and the flasher for the hazards should be mounted on the left side of the dash slightly below the headlight switch. They will be most likely round and made of plastic or glass. Turn on the 4 way flashers and follow the noise. The signal flasher will be close to the 4 way flasher and look similar