Jetflyer,
You're doing a lot of good thinking and many of your scenarios appear to be logical. However, your focus is entirely on the reported heat damage to one of the engines and how that might have happened.
I assume that you currently fly the CRJ so I offer some suggestions.
1) Don't focus exclusively on the possible "hot start". Focus on the entire scenario.
2) Before you get to how a hot start may have developed, think some about why the engines flamed out in the first place, apparently at the same time.
3) Think again about the altitude (FL). Then think about the speed. Then think about the AOA.
4) Then think about this - have you ever noticed the ignition lights come on temporarily during a brisk rotation in a normal takeoff? If so why does that happen?
5) Now go back to #3 and think about the stall protection system in this aircraft. Review your manuals and answer this question for yourself: What is the 1st thing that happens in the stall protection sequence? Why is that 1st and not the shaker?
6) How does the auto ignition system function and why is it there?
7) Go back to #2 again and ask yourself the why question.
Now apply any new information you may have found from those suggestions to your theory. I'm not trying to contradict or shoot down your theory, but only to expand your thinking (which is very good) a little more.
Remember that the NTSB report says the crew initially reported the loss of ONE engine and did not report the loss of a second engine until at 13000 feet. Yet the same report says that BOTH engines flamed out at the same time. Why does there seem to be a contradiction in this?
What would trigger the "memory items" you mention and how does that relate to the difference between what the crew allegedly said and what the NTSB says was a simultaneous double failure? What is the reason for the 1st memory item, given the "logic" of the systems?
Which engine was the one with evidence of heat damage (they didn't say both)?
Was that the same engine that the crew first reported as having lost or was it the other one that they reported at 13,000? Do we know that?
If the crew actually reported one engine out, but in fact both were out, why did that happen? Is it just a coincidence or is the report we got wrong?
Now go back to your own scenario and all the bells, whistles, noise from the ADG, etc., that you outlined and tweak it a little with your new information, expanding on the heat damage question.
We won't know what happened for some time, but your thought process is a good one. The more you know about your airplane the more surprises you can avoid. Even if your theory turns out to be way out of line your thought process might serve you well in the future.
PS. More food for thought ~Do the shaker and the pusher still function if the ADG is the only source of electrical power?