The only thing that makes sense to me based on the information you gave, is that you have either a bad SATA cable for the HDD, or maybe just kind of a loose connection for that cable to the MLB.
Which is actually good news, because that should be a very cheap repair as Apple repairs go. The biggest expense will likely be the labor, as the part will be next to nothing.
I suppose it's also at least possible some other component, like the airport card, has somehow managed to work itself loose, despite two screws holding it in place, just enough to cause some kind of intermittent grounding issue. That seems highly unlikely however, and given the age of the unit, plus the fact that the HDD SATA cable is bent at at least two rather sharp angles internally, that seems a far more plausible source of the issue. If you try the booting from the ODD or even an external HDD, then you should be able to all but prove it's the HDD SATA cable. If you want, you can just remove the internal HDD, put it in an external enclosure, and try booting it that way. Then you can be absolutely sure it's not the HDD, because while highly unlikely, it is at least possible you got a defective drive as your replacement. So slapping that into an external enclosure would allow you to conduct two tests at the same time.
I would also suggest enabling the sudden motion sensor again, if you haven't already. Certainly a valid thing to test, but you have clearly ruled it out as a source of the problem.
Here's a puzzle that I would greatly appreciate any input on:
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