I'm trying to troubleshoot the initial issue where it starts fine, idles fine, but when pressing the accelerator it begins to speed-up, slow-down, speed-up, slow-down in a very rhythmic pattern. This is in P. It will go back to idle fine, and speed up some, but at some point it starts that pattern.

I've cleaned the EGR ports, the ERG valve, and replaced plugs. The CEL is ON - has been ON for sometime. The last reading called out the EGR valve. But I have not pulled codes recently.

One added variable is that now at times it will not engage the starter. I hear a solenoid/relay click, but the starter does not engage. The lights shine bright, so I think the battery is ok. Randomly, after I keep trying to start, the starter will engage and it starts right up.

This all began as we were driving, the engine stopped. We got a jump start (not sure if it was needed), and got it started again and drove 40 miles, but I believe that speed-up, slow-down may have been happening then (wife and daughter were driving). The next day it started right up, but after driving 1/4 mile, it died.

 

F22 engine at hot idle in P or N? Check and clean the FITV and IAC. Most likley the FITV is dirty and the adjustment screw has backed out a bit not closing the air bypass port in the FITV when the engine is hot.
Check the Accord FAQ on how to fix and clean it.