Your bottling bucket actually came with a lid and a hole for an air-lock? If gas is escaping through your air-lock then your lid is probably air-tight and you are fine. It's preferable that way.
I'm guessing you'll need to bottle from your bottling bucket but it's in use. I recommend:
- Clean and Sanitize Everything
- Wait 1 full week (after you pitched your yeast) then move your beer to your carboy by racking your beer (siphon, not the spout)
- Leave in carboy 2 weeks.
- Rack your beer (again, using siphon) from Carboy to Bottling bucket, bottle!
When transferring your beer to bottling bucket you should have already mixed in your priming sugar. The siphon should be set up to cause a swirling motion from the bottom for even mix of priming sugar.
Again, before moving beer always sanitize, sanitize, sanitize. Oh, and sanitize.