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When you water, do you add enough water so that it drains out the bottom every time you water? Elephant ears like moist to wet soil and will grow in several inches of standing water. Depending on the size of your container and your growing medium, it may need lots of water daily. Certainly being too wet is preferable to too dry. Older leaves die off naturally and can be removed. They are also "heavy feeders" and need a regular fertilization.

Elephant ears should grow well in partial shade - 4 hours of direct sunlight daily. Are any of the leaves damaged? I've seen animals sit in the middle of large plants to get out of the sun. The soil needs to be moist at root level. The plant doesn't care if the top 3 inches are moist; it needs moisture where the roots are. I would use a trowel to dig down below the bulb and check moisture at that level. With such large leaves and hot weather, they need lots of water. Checking 6-8 inches down is the only way to know if they have enough.

I suspect that the browning of the leaf edges was not due to bugs, but to water deficit (drought stress). The watering was not sufficient and it made the leaves susceptible to burning from the soap and oil, and the drying continued with browning of the entire leaf surfaces.
All you can do is to water adequately and wait for new growth to appear. Consider using a soil moisture meter to take the guesswork out of water needs.

https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/testing-moisture-in-plants.htm

https://www.domyown.com/luster-leaf-rapitest-mini-moisture-tester-1810-p-7732.html

This is a type of fungal infection. This can be controlled by replacing 1/4 of the water in your container with peroxide when you water next time. This will only take 1 application.

Make sure that you only water when the top 2 inches of soil are completely dry. If it is still moist when you water again, then it can cause this issue.

Although these plants are suited for water, they will not be suited for water in a container. The water will go stagnant and cause an infection.

This article will help you with tips on the care of these plants indoors: https://www.gardeningknowhow.com/ornamental/bulbs/elephant-ear/growing-elephant-ears-indoors.htm

It will not hurt to cut these roots sticking out, however it would be a better idea to transplant these to a bigger pot. Here is an article that will help: https://www.gardeningknowhow.com/ornamental/bulbs/elephant-ear/growing-elephant-ear-plants.htm

Yes, this can be fine. This will definitely keep them from falling over in the future.

When you dig them up to overwinter, or divide them out then you can plant them deeper when it is time to plant. In the meantime, topping off the soil will help.