Very true. However, all could possibly be legitimate emergency procedures.
Wrong - legitimate emergency procedures are different. See below for your detailed arguments why.
I doubt that commercial pilots take off as soon as they reach the minimum ground speed (excluding, of course, the event when they are taking off from shorter runways).
No, you will take off when you pass the rotation speed (Vr). As you always lift-off with full thrust if possible, any difference in the distance needed to reach Vr means a technical problem.
I also doubt that the pilot's intention was to rise quickly, he was probably worried more about getting off the ground so he didn't run off the end of the runway, and keeping sort-of close to the ground, so in the event that the second engine failed, the vertical speed on ground impact could be minimised.
That is no legitimate emergency procedure. That what you describe is called pilot error. After take-off you want two things: Gain altitude and reduce drag. For the first you want a high rate of ascent, for the second you want the right rate of ascent. You don't want to fly NOE even in emergency. You want to get at least to about 3000 ft in a airliner, so you can safely align with the runway again. Too low is always bad, too high sometimes bad.
You can land practically as long as you are well above decision height. But 3000 ft is better for most jets.
Not sure what to say about the reversed thrust, perhaps he was trying to "clear the engine", spitting out whatever foreign body had entered it? I mean, people do strange things when they panic. Or it could have been standard procedure to do such a thing (still on thrust reversal here) in such a situation.
Reverse thrust does not work that way and again, you describe a pilot error - such a operation is not recommended at all for planes, which are not designed for deploying reverse thrust while in air just like the MD-80 is.
Reverse thrust does not invert air flow inside the engine, but instead directs a part of the exhaust or bypass flow in direction of flight. The engine still operates normally at high thrust. No pilot would ever even think about cleaning the engine that way. Also, when an object already destroyed the engine, you will want to shut the engine down as soon as possible, so the risk of further damages to the plane get reduced.
Again: What you imagine as emergency procedure is actually describing massive pilot errors.
The goal of all emergency procedures is NOT to solve the technical problem leading to the emergency but to keep the situation controlled. If a problem can be solved by procedures, it will get solved, if it can't be solved, other priorities are more important. Deploying reverse thrust in flight is leading almost always to a uncontrolled situation and is thus not a emergency procedure at all.
When a engine fails after you passed decision speed (V1), you have only the following option: Lift-off, raise at reduced rate to stay above stall speed and fly around the airport for landing there again. Or at least fly to a nearby airport, if this is easier as a round around the airport (Like the Concorde crew attempted in the Paris crash).
The pilot did very likely not decide to abort the flight after passing V1 - the plane lifted off before the reverse thrust activated. Either the pilot panicked (It is not that easy to activate reverse thrust, you need to overcome a friction lock) or the more likely cause is, that a hydraulic failure deployed the thrust reversal.