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Pot-grown begonia plants can be stored in their containers as long as they remain dry. They should also be relocated to a protected area that’s cool, dark and dry. Some begonias can simply be brought indoors prior to the onset of cold weather for continual growth. This article will help with growing them indoors: https://www.gardeningknowhow.com/ornamental/flowers/begonia/begonias.htm

Most cuttings are taken in spring but plants can also be divided while dormant. For tips on overwintering, here is an article that you may find helpful: https://www.gardeningknowhow.com/ornamental/flowers/begonia/wintering-begonias-overwintering-a-begonia-in-cold-climates.htm

If watered frequently with clean water, it should be ok. The non-salt water will work to leech away any salt that may make its way into the container. Just make sure that you water regularly and generously.

Divide begonias while dormant or when the shoots are still small. Simply dig up the plant clumps and gently pull the corms apart either by hand or you can use a knife or spade shovel. Then replant elsewhere. Here is more information on how to divide plants: https://www.gardeningknowhow.com/garden-how-to/propagation/propgen/dividing-plants.htm

It sounds like it has a fungus. This is caused by poor air circulation and high humidity. With this plant, I would recommend increasing air flow to help the problem. You can either move it somewhere with more air movement or place a fan near it. It will also help to treat it with some fungicide.

It is possible, but the tuberous ones do not take as well to the indoors as the other kinds and need extra light and humidity. These articles will help you:
https://www.gardeningknowhow.com/ornamental/flowers/begonia/begonias.htm
https://www.gardeningknowhow.com/houseplants/hpgen/using-pesticides-and-other-chemicals-on-your-houseplants.htm