I think I know what happened but I want to see if someone else can confirm this diagnosis.
I drove this car home after I bought it pretty much sight unseen.
I've done a LOT of work to it, very little was on the engine.

The car ran great for a few weeks. Suddenly it would have a rough idle at a cold start, all of a sudden it would snap out of it and run perfect.
the last time it ran decent it had the rough start snapped to running good then it went back to a rough assisted idle..
I have to hold my foot on the gas to keep it idling at 400 rpm. It will not try to rev farther like it's in a limp mode.

If I hold the gas in just the right spot it will act like its trying to snap into running normally but it just lasts for that instant.

The person I bought it from drove it several hours to his house with no battery.

 

Fuel filter?

Vacuum leak?

IAC?

What have you checked so far or replaced?

 

IAC is good, No Vac leaks, Fuel filter is a possibility, but not suspected since it happens suddenly and started while it was on jackstands. A new one wouldn't hurt though

 

Spark plugs?
Spark plug wires?
Distributor cap?
leaking oil into spark plugs?

 

I put new seals in the distributor, Honda wires, new NGK plugs. VCG is good
If I pinch the fuel return the car bogs (the little it can) and dies. If it had low fuel pressure it should have improved.

I think the PCM is cooked. It's holding back the injector duration for some reason so it can't make power.

 

Do O2 readings give you any clue? Running lean? lazy readings and not responding?

 

I didn't think of that. I'll give it a shot but it's going to be tough. Whatever it's doing it runs hot in about 10 minutes.

 

Either they are lying, or that is one tough cookie of an alternator. Running with a dead battery is believable, running with no battery would damage the alternator.
Have you checked for any stored codes?
Have you verified the various warning lamps are working at prove out? BATT, CEL, OIL, ABS, SRS?
If the battery is holding a full charge of 12.4-12.6 Volts?
What is the voltage when the engine is on and running?
Have you verified clean tight battery terminal connections?
Verified your ground connections?
G1 at the radiator support and main cable down to the transmission.
G2 engine mount left side bonding jumper and valve cover jumper.
G101 engine harness ground under the boost chamber on the end of the intake plenum.
Verify voltage to the fuel pump.
Verify actual pressure to the injector rail.
Backprobe TPS and MAP sensors on the intake manifold.
Verify TPS voltage output is smooth and linear when opening/closing throttle. .5-4.5V
Verify TPS input voltage is 5V.
Hot running or damaged cooling system will cause the engine to run erratically.
It runs hot as in overheats or runs hot as the temp gauge shows slightly warmer than normal?
Verify the fans kick on when the temp gauge gets warm.
Have you verified the coolant system is properly bled of air?
Engine off.
Open hood.
Remove radiator cap, verify coolant is full.
Verify recovery tank is also full.
Find bleeder on backside top of the thermostat housing.
Open bleeder, ideally attach a clear hose to nipple, watch for air bubbles. Should be a solid stream.
If air bubbles, allow to bleed air, close nipple.
Start engine and allow to warm up normally.

If the engine still overheats
Do a compression check.
What are the numbers?

 

Well, I sorta have to eat my words. It must have jumped time somehow. The timing belt wasn't loose when I pulled the covers, but I did a timing belt and all of the seals and it's running like a champ now.
I've messed up on timing belts before and I've never had one act like that so it's a new one on me.

I did find the cam gear did not have a key in it so the cam gear could have slipped. I didn't look when I pulled it because of course you expect it to be there!
Thanks anyway.