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Yes, once the leaves have turned yellow or brown, you can cut it back.

You can. It will not hurt the plant, but it will not help it either. Pruning the old flowers is done more for aesthetics.

It is either the fertilizer or the sunlight. If you use fertilizer, you may have given them too much nitrogen, which encourages leaves and discourages flowers. If you suspect this may be the case, add phosphorus. Here is more info on phosphorus: https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/phosphorus-plant-growth.htm

A lack of sunlight can happen if a nearby plant has grown to shade the area. Check the area and see if the calla lilies are getting enough sun.

If the bulb is soft, there is nothing you can do, as it has rotted. Healthy bulbs are firm to the touch. It will need to be replaced.

There may be pests on the plant. They can make the leaves "drip". These articles are on the pests that can cause that:
https://www.gardeningknowhow.com/plant-problems/pests/homemade-aphid-control-a-natural-way-to-kill-aphids.htm
https://www.gardeningknowhow.com/plant-problems/pests/scale-bug-how-to-control-plant-scale.htm
https://www.gardeningknowhow.com/plant-problems/pests/mealybugs-white-residue-on-plants-leaves.htm

The should bloom in about 2-3 months, depending on your conditions.