Our Technics SL-1200MK2 has issues with the direct drive. It does not spin at a constant speed. When stopped, it does not stop immediately. What can be done to solve this?

To adjust stopping speed you'll have to remove the plate by pulling it up. Then there is a plastic cover that has some screws, remove the screws and remove the plastic cover, you'll find a variable resistor that reads something like brake. Adjust it to the left or to the right with a screwdriver and see at which position works best.
To solve the constant speed problem you'll have to replace the pitch regulator.
Go to any panasonic/technics service and buy the pitch. If you have a bit of soldering skills you'll be able to replace it by yourself. You can access the pitch regulator unscrewing all the screws under the device and then removing the cover.
Hope you can solve the problem!