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It appears you are on the edge of the zone where elephant ears can survive outside year round. Once your last frost date has passed, you can plant them out with no problem.

This article will help with growing: https://www.gardeningknowhow.com/ornamental/bulbs/elephant-ear/growing-elephant-ear-plants.htm

Leaf curling can be caused by numerous things, including too little or too much water. If yours are in full sun, it could be that the plants are curling up to try to conserve water; therefore, additional water may be needed. Check the soil around the plants to see how dry it is and water as needed to keep the plants hydrated.

Are they in full sun? While they can be grown in sun, these plants prefer partial shade and moist soil conditions. Even though you're watering, the area may not be absorbing it due to evaporation. When the foliage turns brown like this, the plant is often trying to protect itself by going into early dormancy to conserve water. You may want to consider moving them to another, more suitable location.

Yes, a cardboard box (in a cool, dry location) is a good option for these. The use of peat moss and newspaper for storing the bulbs is fine too, as long as it remains dry. It's more of a personal preference.

It sounds like your plant has a fungus. Treat it with a fungicide, and this should clear up the spots.

Not every cutting will take root. It is just the way it is. But, there are some things that you can do to increase your chances of a cutting rooting. It sounds like you are rooting in water, so with that, change the water about once every 1-2 days. Also, try adding a little rooting hormone to the water (available at most places that sell plants). This will help to stimulate rooting as well.

If these cuttings are already dying, they may be too far gone to root, so you may need to get more cuttings to root.