I don't know of any tricks for winding the reg hoses so that you can store the reg set back in its bag. Just wind them and be careful not to kink the hoses. (That's why the reg bag is round--to facilitate this.) I take the reg bag on the plane as a carry-on or "personal item" and check a larger bag that holds my BC, wetsuit, clothes, etc. The BC is pretty tough--it will be fine in your checked bag. In fact, the reg set would probably be fine in your checked bag, too, but many of us have concerns over lost baggage and theft. Some people try to take EVERYTHING as carry-on baggage, but that is a feat.
Lost baggage indeed sucks. On a trip to Bonaire last year my checked bag (BCD, wetsuit, clothes) took a 2-day side trip to Venezuela without me when we changed planes on Curacao. The dive op let me borrow a BC and wetsuit for free (very kind!), but that was two days out of a week that I had to use gear that I was not accustomed to. There are many threads on SB about how to pack dive gear for a trip, including tips for carrying-on more items instead of checking them.