I assume that there is one main pass between Lhasa and Kathmandu. At what times of the year is it most likely to be open? Also are there many passes between China (Tibet) and Nepal?
Yes, Zhangmu / Kodari will never be closed unless during March (for foreigners) when Tibet is closed and permits not issued.
But the road (Highway 318) is usually quite good all year round, there may be some temporary blockage, but it's really not very common. about 5 years ago