Short answer to composting in a 5-gallon bucket -- absolutely.
The worms are surface feeders so would probably prefer larger surface areas. However they do adapt to different containers, temperatures, feed, etc.
Worms don't read rules about vermicomposting but if you print them out and add to a bin they will turn them into great compost.
That said, you will probably have issues with overfeeding. Resist if you can.
Light on food waste, heavy on bedding.

Yes a 5 gal pail is large enough. You could make it into a flow through design where you add scraps at the top and harvest vermicompost from the bottom.

There are a few DIY designs shown at the link below

Here is a link that might be useful: DIY Flow Through bins: a collection of links

I use only smaller than 5 gallon buckets. My worm population is suitable size for my household kitchen waste needs. Still working on flow through actually flowing through. Never thought of having it read the rules about flow throughs.