Do they go dormant, and if so, how long?
They do go semi dormant - don't let them dry out completely even if they have no leaves - they still need some moisture. As long as the rhizome stays firm then it is good to go. If it gets mushy then rot has set in from too much water. If it is dry and disintegrates then it was allowed to dry out beyond its ability to withstand drought.
Full dormancy plants on the other hand need no water for weeks or months. For instance I bring caladiums indoors for winter and let them dry out completely for a few months before waking them up with some water,
Rexes should put out new leaves when conditions are right - light, warmth, water. I have 3 to 4 rexes that are just starting to put out new leaves but then I have dozens more that are quite bushy so there is no hard rules for "rex" begonias. I grow them under shoplights in winter and water 2 to 3 times a week. Your conditions may be different though.