Congratulations on your new-to-you K1200GT! I hope you enjoy it as much as I enjoy mine. Here are the answers to your questions:
* Yes, "clunky" shifting is common to BMW motorcycles, especially the KGTs like your "new" K1200GT. Some people "preload" the shifter by applying light pressure up or down on the gearshift lever before engaging the clutch. With practice, you will get better at shifting, but it will always be a bit clunky, especially when you engage first gear from neutral at a stop.
* You can easily adjust the gearshift lever position with an open end 10mm wrench. There are two 10mm nuts on either end of a short "turnbuckle" - the short rod attached to the gearshift lever. Loosen the two 10mm nuts using the open end 10mm wrench, then rotate the turnbuckle rod with your fingertips. Watch the rubber toe shift part move up and down. Tighten the 10mm nuts when you are satisfied.
Hint: I carry a small 10mm wrench in the glovebox of my K1200GT. Every once in a while on long trips, I feel a need to make a tiny adjustment so that the shifter feels "just right."
Note: You cannot adjust the brake lever position.
* Some guys found a way to make shifting smoother by inserting bearings on the inside of the shift lever. BMW added bearings to the shifter on the K1300GT models. See this thread: