Can't shift into gear while engine running
 

I just replaced the blown slave cylinder, bled the system, and now I can shift thru the gears while the motor is off, but not while on. When attempting 1st, the car will move forward just a tiny bit, and does the same backwards when trying to get it into reverse, but no grinding noises.

Did I most likely not bleed the system correctly, or has something else in the system failed? I used a vacuum bleeding kit on the slave cylinder bleed valve, and topped off the brake reservoir as needed, but the shop manual says they use an Audi tool that also applies pressure to the fluid at the top of the system in the reservoir.

I've checked both ends of the master cylinder and there's no signs of fluid leaking from it, but I'm not sure how they tend to mechanically fail.

Clearly your clutch is dragging. I had one Audi that would drag and be very hard to shift, but didn't grind, so similar to yours. Friend of mine forced it into gear passing uphill on a mountain road, blew the clutch up. Never saw a clutch before or since in so many pieces. Anyway, it could still be your hydraulics, but I have also seen debris on the clutch shaft cause them to drag, the driven disk needs to float off of the flywheel for the clutch to disengage, and if something is in the way preventing that then it is going to drag.

Yikes. Really hoping that's not the case, but thanks for the feedback.

Alldatadiy.com has various repair information that may help with your clutch issue. Good Luck!