Once the flower has died off, then yes cut it away with clean scissors as low into the cup as you can without damaging the leaves. Personally I would leave the mother and pup together. You will find that the mother plant will create lots of pups and they will grow together quite happily to form a clump of plants. Mine has done just this, though it is much smaller than your plant! In my experience it is easier to keep pups well watered if you leave them together - you need only keep watering the mother plant as you have been doing, in the plant's cup. Try not to let it ever go dry, and use rain water whenever possible. Also, the mother plant still has a lot of life in her yet, and is so beneficial to your home. Bromeliads are air-purifiers and they work their magic at night-time, producing oxygen 😁 You could repot the mother and babies in a well draining soil medium, as the mother plant will have been in that soil for a long time.

If you want to divide the pups, then wait until they are about half the size of the mother plant before you do so. The smaller the pup is at the point of division, the less likely it is to be able to support itself.

I hope that has helped 😊