P0300 - living the nightmare
 

New fuel pump, filter, injectors(spider assembly).

New plugs, wires, coil, distributor drive gear, camshaft position sensor, cap and rotor.

New catalytic converter.

No DTCs other than random misfire. The engine runs like a Singer if you throttle it up gently while idling. But if you stomp it and load the engine out it misses badly.

While cranking, it lurches and has torn up 3 starters. It has a hard time cranking, but it will crank and run - smoothly. I can ease into it and get it going, but it bogs down and misses the worst under hard acceleration. Though it continues to random missfire while idling and throughout driving range.

Questions:

Will sticking valves cause RANDOM misfire? Or would that be single cylinder DTCs.

Why is my engine jerking and making a horrible sound when I'm trying to start it and has torn up 3 starters since the problem started?

I had system too rich DTC periodically due to a faulty poppet nozzle for years and it never caused problems. It just suddenly started the random misfire recently and a reputable shop diagnosed it as the spider assembly. I replaced that and all those other parts since then, what problems can be caused by years worth of rich fuel mixture?

What are ALL possible causes for random misfire?

Dumb questions:

Can a weak battery cause fault in my PCM or any sensors? Shouldn't the alternator maintain proper voltage to everything. Cause I have had a weak battery throughout this ordeal.

How important is that little black plastic ball mounted on the bottom of the hood that has a vacuum line running to it?

Black plastic ball
 

That ball is a vacuum reservoir. If it or the hoses leading to it leak, it will cause a lean condition and poor throttle response (like you describe) along with poor A/C vent control, malfunction in the cruise control, etc.
But it wouldn't cause the starter grind.

Run a search on P0300. You should find more than enough information to keep you reading and checking things for a good long while.