Welcome to the herd, Daman.
Your R65 looks pretty tight. And Rob is correct about that "ugly rack"; list it within the Trading Post section and you'll likely sell it fast, if you are so inclined. It is a rare bit of BMW kit.
Get some Simple Green, your girlfriend's toothbrush, some Scotchbrite pads, perhaps some mineral spirits, a cold six-pack and get busy. Avoid steel wool!
I'd counsel that a little bit of original surface patina adds to the Airhead mystique. Clean her up a bit, keep the bike in fine fettle and enjoy.
Drop the float bowls and check the small reservoir that the enrichener tube fits in; the little brass tube you'll see. At the bottom of the reservoir is a tiny "jet" or fuel passage. If it's clogged, no fuel will be drawn through the enrichener circuit and you'll need to hold the throttle open during warmup. Use a single strand of copper from a short piece of 16 gauge multi strand, or so, wire. Probe gently to clear the passage.

Your neutral / clutch issue is the bike asking you for a transmission spline lube. Very important maintenance issue.
Nice jugs and mugs ya got there [smiley=beer.gif]
Monte