Firstly, fix the issue that is causing the charge light to stay on with the new alternator. May be unrelated, but definitely worth doing.
Your battery sounds like it could be on its way out (running them flat at all damages their life) but won't be causing the main issues.
The main problem seems like it could be an intermittent short somewhere. I'd be looking somewhere in the igntion system as the tacho dropping to zero, while the dash lights are rock solid suggests a missing signal. Check all the wiring on the ignition side. It could be a loose pin/plug, a wire with worn insulation that rubs shorts out randomly, or a bunch of other things. With the car idling try wiggling all connections and wires to see if you can recreate the problem.
Intermittent issues are a pain to track down, but work through it methodically and you will find it.