People use carboys because they can fill it up to the neck and leave very little room for air, since the beer is no longer fermenting and producing co2.
The general story with this is that carboys are impermeable to oxygen while buckets are not. This might possibly affect your beer if you are storing it in the container for a long period.
No. As a matter of fact, the only time I use a carboy is when I have to age a beer for a long period of time. The buckets are technically o2 permiable and eventually (over months) your beer can oxidize. Other than that, buckets are superior in every way.
You need an airlock, unless the beer has been finished for a very long time and no longer off-gassing at all, but even then changes in temperature or barometric pressure can mean some pressure in the carboy. Even if I have a wine in a carboy for a year, it has an airlock. I just have to remember to add water (due to evaporation) from time to time.