you can stick in an sw20 mr2 3sgte, as far as i can gather.
as for sticking in the mr2 gearbox thats a bit more difficult. if you did do such a conversion dont buy the engine then the loom then ecu etc etc - it'll end up costing you a mint. if you end up taking this route i would suggest you find an SW20 halfcut (which will have the loom ecu etc etc and a bumch of other stuff you can use in the swap) and save yourself the grief.
your catch in that instance will be that the Mr2's gearbox has its gearshift linkages on (what i will call) 'front side' of the engine in east west fitment. so all the linkages enter the box from the front of the car, from the shift lever in the cabin thats in front of engine.
in a corolla or other fwd car the linkages exit the gearbox from the REAR side of the box, and go *backwards* in the car to the shifter in the cabin (BEHIND the engine). obviously thats a problem.
this isnt a dealkiller it is possible to drill out the opposite side and sort the linkages that way (ive seen someone - cant remember where tho - detailing how they nicked a fwd corolla box, drilled the box to accept linkages from the other side and then mounted it to an MR2. in this case youd need to do the reverse).
as i say not a dealkiller but its an expense you probably can do without (and fabrication type labour is the expensive stuff) so something therefore to keep in mind. not as expensive as the fab work sorting 3sgte mounts into a corolla but enough to think it through first, id say.
ive not commented on allwheeldrive/alltrac stuffs as i know nothing about those. sorry!
good luck whatever swap you end up picking!
Cheers