The KKH has bits that close from time to time due to bad weather, bridges collapsing or political protest, but is an all-year-round road.
I wouldn't go up there in the winter on a bike though. It is cold.
The road was built to cement ties - literally - between Pakistan and China, so yes, you can drive right over the border. The paperwork/permission is the problem.
There is plenty on this site about getting into China. Have a look around. In a nutshell: basically possible, but very costly.
Are you are looking for a Oz to Europe overland route Stefan? Sadly it is not possible through south asia, as you have to ship the bike over/around Burma. If you want to keep your wheels on the ground then Russia is your choice. I can put you in touch with another Queeslander who did this very route as it happens. PM me.
Around the world 2000-2004, on a 1993 Honda Transalp