I took the beetle out for a test drive today. The fixes I made to the gear shifter made a huge difference. The shifting is much more precise and it doesn't rattle while it's in gear. Yay! The steering didn't make as much of an improvement. The steering doesn't wander as much, but the vibration at 45 mph is still there.
Since I had the gear shift out anyway
I replaced the shifter block-off plate. The old one is on the right, and the new one on the left:
You have to push the gear shift down to get the beetle to go into reverse. The shoulder at the left of the hole in the block-off plate is what holds the shifter down and keeps it from popping out of reverse. The shoulder on the plate on the right is worn:
Before, my car had slipped out of reverse from time to time, so I thought maybe replacing the plate would fix it. It's just fine now.