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My neighbors very large cactus was hit and broke off a large chunk. The bottom part is old and wooden-like. Would I be able to root it? If yes, since the older cactus is wooden, do I have to wait for the stump to harden anymore before I stick it in dirt?

I say stick it in without waiting.

It looks like the top half of the older pad is pretty green. That's where you'll have the most success in rooting. Actually, you could just lay the pad flat on top of soil, as you have it shown in the first photo, and it should send out roots from several of the areoles.