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Cut down flower stems on daylilies once blooms are faded. Daisies will produce more blooms if faded flowers are cut back to the next visible bud. On tall shasta daisies, this may be 6-9 inches below the bloom. Look closely for new growth where a leaf meets the stem. Flower stems may be cut back to basal foliage after blooming is over. The main foliage is still needed to prepare the roots for winter so it should remain intact until it yellows in fall.

You can cut off the dead blooms. This will not harm anything at all. This article will give you more information on caring for daylilies: https://www.gardeningknowhow.com/ornamental/bulbs/daylily/growing-daylilies.htm

The smaller roots will not produce any plants, However you can divide the larger rhizome into multiple plants. This hair-like feathering on the roots is caused by its relationship the the good bacteria in your soil called mycorrhizae, Without these the plant would not thrive. Here is a link for more information on dividing daylilies: https://www.gardeningknowhow.com/ornamental/bulbs/lily/dividing-lily-plants.htm

This article will help you.

https://www.gardeningknowhow.com/ornamental/bulbs/daylily/daylily-seed-propagation.htm

If a daylily is allowed to set seed and the seeds are not removed then they can fall to the ground and germinate into a new daylily plant. That plant will not be the same as its parent but may be quite plain. It may also be quite vigorous and outgrow its parent as time passes. If that is the case then after a few years only the seedling will be growing and flowering and then some may look at it and consider that it looks old-fashioned and like an ancestral daylily. But the original plant did not revert; it was more or less killed by its offspring and the offspring was not a parent or ancestral daylily.

Spread several inches of moist peat moss in the bottom of a cardboard box or plastic tub.
Lay the rhizomes on top of the peat moss, placing them so they don't touch each other. Rot can spread from plant to plant if the rhizomes are touching and one rots. Alternate layers of peat moss and rhizomes until the box is full or you have placed all of your rhizomes. End with a layer of peat moss.
Store the box in a cold, dry place where the temperature is 35 to 45 degrees Fahrenheit.