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As long as this is in container, or it does not stay below freezing for very long, then it will stay alive just fine. If it does get below freezing in your area, then you will want to move this indoors on those nights. They will start to show damage around 40F degrees or so, but will not hurt them.

This article will give you more information on the care of Begonias over winter: https://www.gardeningknowhow.com/ornamental/flowers/begonia/wintering-begonias-overwintering-a-begonia-in-cold-climates.htm

This is great news. The Destiny series are tuberous begonias. (Wax begonias do not make tubers.) You can overwinter them as houseplants but the most common approach is to unearth the tuber, allow the plant to dry for 5-7 days, shake off as much soil as possible, and cut off the stems flush with the tuber. I confess that I cut off the foliage before drying the tuber and this works for me. When the stem cuts look dry, dust them with garden sulphur and store the tubers in dry peat moss or wrap them in newspaper and hang in a cool, dry place. The mesh bags from onions and oranges are good for hanging bulbs and tubers. 40 F is the ideal temp to aim for but most of us are closer to 55-60. Avoid freezing temps. It takes them awhile to get going in spring so you might want to pot them up inside, water and place them where it is warm and sunny. They should not be put into the garden until all danger of frost is past and night-time temps are above 50. They sulk in cold soil and are at risk of rotting. Please read the following article. https://www.gardeningknowhow.com/ornamental/flowers/begonia/wintering-begonias-overwintering-a-begonia-in-cold-climates.htm

It is, indeed, some type of Begonia. It is hard to tell which, because of its condition. The second plant is a Pothos! This is a very hardy and forgiving plant. These two articles will help:

https://www.gardeningknowhow.com/ornamental/flowers/begonia/begonias.htm

https://www.gardeningknowhow.com/houseplants/pothos/pothos-plant-care.htm

Most of the time, this is a natural occurrence with age of the plant. Sometimes it can be an indication that they need more light. This article will explain what's happening in more detail: https://www.gardeningknowhow.com/plant-problems/environmental/fixing-color-faded-flowers.htm

This collection of articles will give you more information on the care of Begonias: https://www.gardeningknowhow.com/ornamental/flowers/begonia/

Are you referring to the propagation dome? Or a heat mat? You can leave a dome on as long as you leave the ventilation holes open. This will help keep the humidity up, while allowing airflow.

They can take a bright but indirect sunlight. Look for dappled sun locations.
https://www.gardeningknowhow.com/ornamental/flowers/begonia/begonias.htm