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These are a unique cross between an angelwing and a tuberose begonia. They will do quite well in a sunny window, needing a little more light indoors than out. These also can be pruned like an angelwing. This can be done anytime indoors. Just prune to your desired height and save any branch tips to put in water on your windowsill. These will root very easily. Just change the water out every few days.

This can be cared for like an angelwing, so this article will give you more information:

https://www.gardeningknowhow.com/ornamental/flowers/begonia/angel-wing-begonia-care.htm

https://www.gardeningknowhow.com/ornamental/flowers/begonia/wintering-begonias-overwintering-a-begonia-in-cold-climates.htm

I love Bonfire begonia. Since it is newly rooted, I would pot it up in potting soil and keep it inside over the winter near a bright window. Let soil dry out well between waterings. Pinch tips of new growth to encourage it to branch out. In spring after threat of frost has passed, move it outside.

You can pot it up very gently. You will need to keep it inside over the winter. Here are instructions for taking care of begonias:
https://www.gardeningknowhow.com/ornamental/flowers/begonia/annual-begonia-care.htm

That is a beautiful begonia. Was it exposed to freezing temperatures? Your description of the plant sounds like what begonias look like after a hard freeze. If not a freeze, your house is rather cool for a begonia. They like about 70 degrees F (21 C) during the day and 60 degrees F (16 C) at night. Consistently cool temperatures can weaken or kill a begonia. If that is the case, you won't be able to revive it. The other possibility could be root rot. If that is the case, sometimes the plant can be saved by carefully cutting away the damaged roots and repotting in fresh potting soil. Here is more:
https://www.gardeningknowhow.com/plant-problems/disease/treating-root-rot-gardening-tips-for-housplants.htm
https://www.gardeningknowhow.com/ornamental/flowers/begonia/begonias.htm

You can take cuttings at any time, if you are planing to root these indoors. Just make sure that the plants are not flowering when you take the clones, as flowering hormone tends to inhibit rooting hormone.

If you intend to root these cuttings outdoors, then make sure that you take the cuttings during the rainy season to ensure that they receive water to strike.

This article will help: https://www.gardeningknowhow.com/ornamental/flowers/begonia/begonia-propagation.htm

There are many Begonia experts here in the states. A Google search will reveal some of the most popular. There you can find email, contact information, and web pages that will help you find a group to trade Begonias with.