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They can be divided in spring or fall.

I dug a bunch of them out from the roots and a good couple of shovels of the dirt out and gave them to a neighbor. She loves them and I don't have to deal with them any more!!

In order to prevent damage to your plants, it would be better to wait until the end of the growing season (probably some time in fall) before raising the height of your beds.

Yes, many varieties of lily are quite hardy and will grow fine in your region. Daylilies, especially, are very hardy plants. Here is more information on growing these lilies: https://www.gardeningknowhow.com/ornamental/bulbs/daylily/growing-daylilies.htm

Most lily varieties do not rebloom, though there are some types of lilies that do. These would be listed as reblooming. You can, however, remove the seed pods and try your hand at planting the seeds, though they will take much longer growing them with this method then by division.

It sounds like your daylilies are "scape blasting". This article will explain more:
http://www.daylilies.org/ahs_dictionary/scapeblasting.html