飞 and 飞行 - Colloquial vs Literary. You use 飞行 during announcements (airports and such) or literary works. 飞 is how you would say it when speaking with people in a non-formal setting. 饺子 and 水饺 - 水饺 is a type of 饺子 (the one which you boil in water). You also have other types, such as 煎饺 for example.

- Alexander Krasnov

I think 飞行 is used as a noun while 飞 is used as a verb. For example, 这趟航班的飞行时长是一个小时。The duration of this flight is 1 hour. 鸟儿会飞。Birds can fly.

- Yen Lui