and on my 87 300e the windows will work for a while after the ignition is switched off, almost as if there's some kind of electrical reservoir
I have yet to check if this is related to the lock doors/automatic window up trick
however, I did have a problem recently where the windows would only work when one of the doors was open, engine running or not. not ideal, particularly when your AC isn't working either...
A search on this forum revealed that there's a diode hidden below the fuse box that controls this and it often blows. sure enough, it had. replacing it requires a trip to your local electronics store for a new diode which you solder in yourself.
this may be somehow related to your situation, if so hope it helps