No you do not have to mulch your daylilies. Most gardeners do to cut down on weeding and watering.
You can put it around but not under
Rabbits and deer eat daylilies. Even cats will chew on daylilies.
There are daylilies in most temperate locations.
Yes. Daylilies can be grown in pots.
Any organic material will do just make sure the ground is well watered before you put the mulch on.
Daylilies are very hardy. The pests that bother daylilies are rust, thrips, spider mites, snails, and crown rot.
Most likely it is because your daylilies have too little or too much water. If you are concerned by the soil, have it tested and ask the expert what amendments you need for your daylilies. Most daylilies are very hardy.
Iris is toxic to cattle but daylilies are not toxic.
Most daylilies are shipped bare root. Generally they are wrapped in damp newspaper to keep them moist from shipping. Because they are so hardy, just put the roots or tubers in water for a day before planting them in your yard.
Use a spray bee killer for the particular type of bees in the mulch bed. Be careful walking around the mulch as the bees may be inside the mulch as well.
Daylilies are not poisonous. In fact, all parts of the daylily are edible. The Chinese use the buds in stirfry. And you may have to spray the daylilies with repellant to keep deer and rabbits from chewing on them.
The best way to mulch a tree is by using a product called Mulch Mat, it is made by Perm-A-Mulch. You get the size you need, lay it around the tree, it lasts for 10 years.