You can definitely separate the pups when repotting, just pull them apart carefully. They may already have some of their own roots which will help them along well. If there are any obvious wounds in the plant or any of the roots were broken off, clean as much remaining soil off them as possible, keep them drive and give them a few days to heal over before sticking the plants in new soil. If the pups have no roots at all keep them dry for a good while - no roots means that water really only encourages rot...
When to repot: well, here in Arizona I only hold off if I know there is 'cold' weather in the forecast, but other than that I usually go ahead and do it whenever, especially if just sticking the plant in a bigger pot and adding new soil. If I know I need to mess with the roots - because of separating pups or worry about some kind of disease I tend to hold off when the plant is dormant and/or night time lows are low <50F.
I am sure there are others that can provide you their insight.
As to the ID: I think this is a similar plant to the Haworthia post in the Plant ID forum so I would suggest some kind of Gasteraloe (Gasteria x Aloe hybrid) or maybe an Alworthia (Aloe x Haworthia hybrid).... but I suspect the former.
