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They can be easily lifted with a garden fork or spade shovel prior to the complete die out of their foliage. Break up the clumps and separate the bulbs, allowing them to dry out some before storing, usually about a week or two in a cool, dry area. Then, clip off foliage, shake off any remaining soil and pack the bulbs in dry peat moss or wood shavings inside a brown paper bag or cardboard box. Place them in a dark area with cool room temperatures, like a basement, until spring.

Just go ahead and plant your cutting in a potting mix with good drainage - a cactus mix with perlite added 1/2 & 1/2 would be fine. Put the pot where it will receive several hours a day of sun; you can use electric light to supplement if you need to. Keep the humidity up as high as possible with pebble trays, and lots of other plants. This article has more information: https://www.gardeningknowhow.com/ornamental/flowers/begonia/begonias.htm

It doesn't have to be warm, but the temp should stay above 45F. Begonias are tropical plants. Here's a resource you might find interesting: https://www.gardeningknowhow.com/ornamental/flowers/begonia/

These are normally available at most garden centers and plant nurseries in the spring. We do not sell them, but can help you with information on growing these plants: https://www.gardeningknowhow.com/ornamental/flowers/begonia/annual-begonia-care.htm

You may think your plant has good light, but compared to what it was used to on the front porch, it's now in semi-darkness. That's probably the major part of its problem. Secondly, it will be using much less water (because it has much less light,) so you may be keeping the soil too wet. Begonias don't like wet soil. Before watering, test the soil moisture. A moisture meter should read around 3/4 of the way between dry and moist. Here's more information about testing soil moisture: https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/testing-moisture-in-plants.htm
In the future, if you take a few weeks to acclimate the plant to lower light, you'll probably have fewer problems. https://www.gardeningknowhow.com/houseplants/hpgen/acclimate-plants-indoors-winter.htm

You'll only need to water, lightly, if the planting medium gets dry, so you should feel it with your fingers daily. If it gets to the point where medium pinched between your fingers doesn't stick together, you can spray it gently with water from a spray bottle. No way to tell you a "how often" rule, it depends on so many variables. Here's an article on growing begonias that you might find useful: https://www.gardeningknowhow.com/ornamental/flowers/begonia/growing-begonias-from-corms.htm