I cut the inflorescence off after it dries completely.

The pup should be about 1/2 to 2/3rds size of the mother before separating, and not smaller. I just break it off with my hands and then plant the bottom in a small pot of soil (fast-draining planting mix and perlite mixture). Yes, the pup will flower too. While the mother plant won't flower anymore, you can hang onto it as it can produce more pups. Eventually, it will die and you can throw it out.

It kinda depends on what genera bromeliad you have. Some its better to leave the pups on and gradually cut away the declining mom. The pups will cover over where she was. With Guzmania leave the pups. Neo's and others they are easier to remove when they have developed their own set of side roots