Do they need fertilization? I know they like well drained, rich soil.
You can fertilize when they are actively growing, an all purpose fertilizer can be used.
Don't over fertilize, you can inhibit the flowering! Moderation!
https://www.gardeningknowhow.com/ornamental/flowers/hellebore/growing-hellebores.htm
When is the best time to move a helebores from one part of garden to another?
If you need to move a Hellebore, it is best to do so in September or October.
I recently planted some heleborus plants and find that instead of growing upright they are falling over to the sides. Is this normal--or what should I do?
Many gardeners tie up or use round supports for their Hellebore plants.
https://www.gardeningknowhow.com/ornamental/flowers/hellebore/growing-hellebores.htm
After flowering, should I cut back the flowers or the plant itself?
The leaves should be left on the plant through the summer and fall. The plant needs them to store energy for next year. You can optionally cut the flower stalks down.
You can trim the old leaves from the previous year in late winter/early spring each year. This improves the plant's appearance in time for the new leaves and flowers to grow.
I live in MT, zone 4. I need to move 3 Lenten Rose plants that are now in a sunny location due to the removal of a tree. When do I move them?
It is best to move them in fall. This article should help
https://www.gardeningknowhow.com/ornamental/flowers/hellebore/dividing-hellebore-plants.htm
If I plant a fancy hybrid Hellebore among my plain normal variety, will it revert to the plain flower eventually?
The hybrid Hellebore you plant will not revert to the plain variety. Since these are perennials, you will be enjoying the hybrid flowers for years as long as the plant is well cared for. However, the seeds of the hybrid variety may produce plain flowers (or other variations) if planted.
What fertilizer do I use on Hellebores?
Just about any all purpose fertilizer will do, but make sure whatever you use has less nitrogen than potassium and phosphorus.
This article will give you more information on their care:https://www.gardeningknowhow.com/ornamental/flowers/hellebore/growing-hellebores.htm