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It sounds like you are overwatering them. They like to be well watered but do not like standing water. Make sure that they are not in standing water. This article may help: https://www.gardeningknowhow.com/ornamental/bulbs/calla-lily/growing-calla-lilies-and-care-of-calla-lilies.htm

Yes, they are but in certain areas, such as cold regions, they may require lifting or additional protection over winter. These articles will help:
https://www.gardeningknowhow.com/ornamental/bulbs/calla-lily/calla-lily-winter-care.htm
https://www.gardeningknowhow.com/ornamental/bulbs/calla-lily/growing-calla-lilies-and-care-of-calla-lilies.htm

This is actually more common than most people are aware. There's no need to worry about the green color. These flowers do that as they mature and begin producing seeds. You can cut them off or leave them be. As for the leaves, this could be attributed to the natural dying down of the plant foliage as it produces seeds.

You can remove the entire stem.

Yes. You can cut back flowers after they die and prune the withered stems as well.

Most likely, it will have trouble blooming this year after dividing.