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Yes, bromeliads can be grown from seed if the flower has been pollinated. Propagation from pups tends to have a high success rate, while growing from seed may have a lower success rate, depending on the species. The number of pups varies by species, but it sounds like your bromeliad is very healthy if it has so many pups.

Here are some articles on how to grow bromeliads from seed:

http://www.bromeliads.info/starting-bromeliads-from-seed/
http://www.bsi.org/brom_info/growing/seed-ja.html

It is very common to have multiple pups from a single Bromeliad. You will generally end up with more then you started with.
It's fun to keep them or gift them to another garden friend.

Let the plant die back naturally and then nurture the next generation.

It sounds like your plant is nearing the end of it's life cycle.

This article has more information for you.

https://www.gardeningknowhow.com/houseplants/bromeliad/growing-bromeliad-plants.htm

No, they are started from seed.

This article will refresh you on the care requirements.

https://www.gardeningknowhow.com/houseplants/bromeliad/growing-bromeliad-plants.htm

Your Bromeliad has reached the end of it's life cycle.

If your plant produced a 'pup', you can remove it and place it in a pot.
Here is a link to refresh you on this plants care requirements.

https://www.gardeningknowhow.com/houseplants/bromeliad/growing-bromeliad-plants.htm