For the primary fermentation, extra airspace would be ideal as yeast need oxygen at first to get off to a good start. Some people even shake the carboy periodically for the first 3 days to get air incorporated into the must.
For the secondary fermentation you would not want to expose your must to air at all. So racking into one of those 3 gallon carboys you bought would be best, you'd want as little headspace as possible at this point.
My wife and I share our plastic buckets and glass carboys for beer (me) and mead (her). We've been told that a downside of this would be that we'd be flavoring each others batches from sharing the plastic buckets...but we haven't noticed any off flavors at all. So go for it with the plastic buckets, they're easier to clean too.
It is a good idea to keep your brews away from light, it could possibly produce off flavors and the yeasties like it dark and cozy anyway. My wife puts one of her shirts on her carboys to keep out light...that way she can peek. I just put mine back into the box the carboy came in.
Some good reading on mead can be found on the gotmead forums...
Good luck and happy brewing.