After a search on line for those names, it appears you have a Rieger (Hiemalis) begonia. Most of these are treated as annuals but you can overwinter them with limited success (in my limited experience with these).

I would only trim leaves that are oviously bad but it shouldn't hurt to trim away leaves you find objectionable (as long as you keep a majority of leaves for growth).

I think if you give them plenty of light then you may be able to keep them going. The few I've tried always kick the bucket sooner or later and I believe it is due to inadequate light and watering practices. They are great growers in summer but lackluster in winter under lights so it is not a huge deal if they go to the compost heap after fall.

Here was mine from last year. It held on most of the winter but finally succumbed in March. R.I.P.

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