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So last night I was driving my Cub Cadet 3235 in the dark. I was going at a pretty good clip when SLAM I ran head on into the cast iron septic vent pipe which stopped the tractor dead in its tracks and dented the deck pretty bad too. I was able to back the tractor out from the pipe, but then when I tried to go forward again, the tractor just jerked a bit and made some clunking noises, but otherwise wouldn't move- not in fwd or reverse. Engine runs fine, PTO engages just fine, but the tractor won't move in either direction. What did I do???? Any ideas how to diagnose and fix this, or should I just take it to a professional repair shop?

This sound bad.
Just to check I am talking about the right mower
Does yours have the horizontal drive V twin engine and a transmission hump from the engine to the transmission ?
If so it drives off a spigot that is bolted to the flywheel with a flexible coupling
These can break off in this type of collision.
Get some one to start the mower while you watch the shaft from underneath,
It should spin at engine speed.
If it looks slow stop the mower and try to turn it with your hands.
You should be able to turn the engine over by rotating this shaft
If not then the coupling ( or shaft ) is gone.
part 46 in this drawing Cub drive shaft
And there is a similar one at the other end.
They come off by unbolting the rear one first then loosening the engine mounting bolts enough to slide or tilt the engine enough for you to get past the cover bellows.
An easier job for gynecologists or proctologists

check out driveshaft for being broken.

Looking underneath while it's idling, I can see that the plastic fan immediately in front of the rear differential is spinning as it should. I assume this means the drive shaft is okay, at least from the engine back to that point. I will also say when the fwd or reverse pedals are depressed, I hear noises that seem to be coming from the differential, like something is slipping. I don't know if that is a better or worse diagnosis than a busted shaft coupling, but again, it seems like the shaft is spinning at engine speed.

Well you have just gotten to the limits of my knowledge on the 3000's
The site that hosted the drawing I linked to does have a breakdown of the tranny.
Trannys are not all that difficult to work on but you must be very strict with the cleaning.
On the 200's that I have done, after repair I did 3 oil changes.
One after 10 minutes one after the test drive ( about an hour or so ) and again at the scheduled end of year service.
The mower is still going strong, but it was jus a broken axel.

Personally I'd evaluate the deck first. If the deck is bent/warped it will cut like crap. If the deck is crap it doesn't matter much if it drives unless you're just going to pull a cart with it or something.

Here is something to try. Put on the parking brake, this assumes that the brake are in working order. NOw try moving the mower. IF it moves then there is something broken in the mechanical portion of the transaxle assembly. Your unit actually has a mechanical unit and a separate hydrostatic unit.

Many of these have a release lever somewhere in the rear to allow you to push the machine around, might try to release and re-engage that device. Usually a coupling sleeve inside the transaxle that allows this to work and that could be broken if the above doesn't do anything.