I'm not quite sure how it works with the cyclopropenyl anion, but as it has as good as none conformational flexibility, I suspect there are two opposing forces at work: hybridization to sp3 to get rid of the anti-aromaticity and reduce bond strain, and hybridization to sp2 to stabilize the carbanion. No matter what happens, you're still stuck with something that cannot possibly be stable in any way.
As for the cycloheptatrienyl anion, a quick google search revealed that its pKa is about 36, making it almost as strong an acid as 1,4-pentadiene. This indicates that the anion is most certainly not flat and not fully conjugated. While it is isoelectronic with cyclooctatetraene, its pKa value makes me suspect that it is partially conjugated and does not share COT's tub-shaped conformation. Beware, however, this is my conjecture.
However, to reiterate your question, rehybridization of a carbanion from sp2 to sp3 imparts a significant energetic penalty due to the increased p character of the orbital involved, and unless some other stabilization mechanism is at work, it is highly unlikely to happen.