I've only replaced one clutch, and I did it with both the motor and the tranny in the car, just cracked and wedged apart.
The wheels spinning opposite directions when in the air is normal. Actually, if a normal MTX car is running in neutral in the air, the tires will spin opposite directions on their own (or just one spins). Just part of regular open differential operation. ( I've also heard of some people whose cars actually creep away on them in neutral

time to change the fluid)
The only thing I could think of is the clutch fork, but since you can't shift into ANY gears, running or not, (clutch in or not?) I'd have to think the tranny is shot. Maybe when the clutch exploded the sudden relase of stress in the tranny caused something to go awry.
The gears in an MTX are always meshing. The gearsets 1-5 are always mating. What changes is which ones get locked to the "output shaft" (or is it input? either way mr. politically correct over there urks me too) So if the selector in the tranny is shot, thats what would cause you to not be able to shift at all.
Starter puts out amazing torque. Its also a teeny gear on a big fat one to make more torque. When you press the brakes when your car is in gear and you try and start it (which should never happen, but you seem to have tricked the system) The starter will either moan in terrible pain or burn your clutch. If you don't have brakes on, the car runs away on you. you said the car tried to move, right? so maybe the whole system is working to some degree. You said the tires moved when you tried to start it in the air, so the tranny can't be seized in any way...
I'm confused as to why, when it starts in nuetral it takes off and when you jammed it into 2nd it still takes off? You can't un-jam it from seccond? I now think it must be something in the gear selecting assembly. You must have always been in gear and now jamming it in, you might have broken something inside the tranny. If you managed to be in 2 gears at once, that would also cause a bind and nothing would move....
Does the shifter do the normal pattern? Does it move side to side in neutral?
Sorry I'm not more useful. Just chucking some more stuff out there.