Most tarvel agents can book the mountain flight. It will be cheaper to book through at ravel agent than to book through the hotel.

Nagarkot is roughly an hours drive and good to see the sunrise etc. There is good resort there called Club Himalaya to spend the night. It may make more sense to spend either the first or the last night in Nagarkot and go there directly from the airport or vice versa. At present I am in india and plan to return to Nepal around the 20th of March.