Megasquirt tuning problems

First, where is your tune file and logs of the issue using that exact tune? We can't diagnose issues as effectively without data.

You have two primary issues—one is possible hardware issues, such as the "rolling idle" with the OEM ECM, or your weird O2 data. That indicates an existing problem that you are trying to cover with MS, or affecting MS. Second, as already mentioned; your tuning method is not sound, and is running you into the ditch. Not to sound demeaning, but it is the equivalent of banging on piano keys and the resulting noise it makes. Easy fix - you simply need to ensure everything is functioning correctly, and follow some simple guidelines to begin making music. Fair warning: if going caburetor, you would face the same or similar issues with harware and tuning methods. You can't avoid it. Sure it would "run", be we assume you are doing this to help it run better.

One is to make changes one-at-a-time so you can see how the engine responds to each change. This is how you learn what each adjustment does, so the next time you see certain symptoms, you have some idea what setting will affect it and in what ways. Another is to use goals in your tuning in various operating areas, e.g., max torque, minimum emissions, max economy, stable and responsive idle, etc. Study what factors apply to reach those goals and begin applying them. We never tune to AFRs or timing—we find the best ones to use together.

What you will find is that we are performing diagnostic tuning. Adjusting settings and values to published, or rumored values (i.e., 14°BTC or 13:1 AFR) is not tuning. Finding the best or most efficient settings or values for operation in various conditions, by testing, using engine feedback and response is tuning. More or less fuel, more or less timing, under the same conditions for data comparison for what is best. That's learning to play this piano.

One more. Tuning for emissions using the venerable 14.7:1 is not efficient, powerful or stable. It is actually de-tuning to reach some specifications rather than performance or efficiency. I strongly suggest you tune for best operation first as a benchmark of known condition, and only then deviate from highest efficiency to set emissions targets and results. Much easier and quicker that way, IMO.
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The MS PNP is hit or miss. It was wonky on my 91 turbo car. Make sure you set the auto tuning of the A/F at rpm out of the idle range.

above link is great help! Once you get idle close you can use timing cells to help make it smoother(the cell below idle rpm a degree or 2 higher)
I had a lot of issues getting mine dialed in. Started out with someone elses timing map.....big mistake. Made a new one myself bases off ADL timing +4 degrees across the board. Also the accel enrichment was jacked as it added 300% fuel everytime you hit the gas all the way to 4k(huge tip in stumble) diy autotune wasn't any help and had to figure it out on my own. Didn't find BlackFoxtuning until later.
Just stick with it and make 1 change at the time!
Hey bro. Are you still having issues with this? Let me know. I can walk you through some stuff remotely or over the phone. I know how you feel. I started with Megasquirt back in 2014 when you had to build the kits yourself and now I have my procharged 416 LS3 running pretty perfectly on a MAF. It took a lot of time and understanding tuning principals and how each change affects the engine. the Idle issue should be pretty easy to fix.