Is it absolutely necesary to do the primary fermentation in a bucket, or can it be done in a carboy? What size carboy is best, 5 or 6 gallon?
I have my first wine kit working, so take what I say with the understanding I'm a beginner, but:
The kits seem aimed at six gallons so go with six on the size. I used a six gallon primary, racked to a six gallon secondary, and then topped off the secondary. The wine kits and online seem to always suggest topping it off to get the carboy full.
One other thing you need with wine making that I didn't have is something to degas the wine -- basically stir the snot out of it. The suggestion from the kit is a long-handled spoon but we didn't have a three-foot-handled spoon lying around. I used a racking cane but plan to get a "lees stirrer" prior to the next racking. Also, based on reading, ignore the time-frame of the kit and go longer in primary, secondary, and in bottles.
Edit: I plan to rack to a tertiary in a couple of weeks, degas and clarify, then wait three or four more weeks to bottle. Current time-line is in my signature below.