what can I do to get them to open? I leave the bulbs in the ground year round.
These are known for being a bit finicky at flowering. Giving them phosphorus and potassium can help with the bloom process, but it is no guarantee.
They can go as much as 3 years in between blooms, so there may not be much that you can do outside of proper care. These articles will help:
https://www.gardeningknowhow.com/ornamental/bulbs/paperwhites/growing-paperwhites-outdoors.htm
https://www.gardeningknowhow.com/ornamental/bulbs/paperwhites/paperwhites-reblooming.htm
How to care my paper white?
Unfortunately, your photos did not come through. I am unable to assess the damage. It could be a number of things, and these plants can be a bit particular. In the meantime, these articles will help:
https://www.gardeningknowhow.com/ornamental/bulbs/paperwhites/growing-paperwhites-outdoors.htm
Testing soil pH and nutrient content, as well as testing for disease may be necessary:
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/testing-moisture-in-plants.htm
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/testing-soil.htm
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/soil-ph-plants.htm
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/test-soil-for-pests-and-diseases.htm
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/making-sure-soil-drains-well.htm
This collection of articles will help you to grow Paperwhites:
https://www.gardeningknowhow.com/ornamental/bulbs/paperwhites
is one whose original sprout is rotten. It is pushing out roots. Should I cut off the dead sprout? All the bulbs had an whiteish bud/stem out prior to planting. All of these original stems are green and thriving with the exception of the one stem that is dead. However, that bulb is alive and pushing out roots. Do I cut the dead stem off? Will it push out another?
This sounds like an attempt by the plant to stay alive! There is no harm in attempting to save it. Make sure to cut off any dying materials, and treat with a fungicide.
I would reduce the water level as well. You want to keep the base of those bulbs out of water, so that the roots have to search a little, and to keep it from rotting.
https://www.gardeningknowhow.com/garden-how-to/info/using-fungicides-in-garden.htm
https://www.gardeningknowhow.com/ornamental/bulbs/paperwhites/forcing-paperwhite-bulbs.htm
https://www.gardeningknowhow.com/ornamental/bulbs/paperwhites