The first one is 'Pink Diamonds' and the second may be 'Peace' (if it was large I would say 'Mikado'). To keep them you might try cutting a couple leaves off each one and propagating them to have some back up insurance (that is what I do now after losing bunches over the years).

I don't know the ID of this begonia but it is similar to one that I have. I find this type of stem, I call it hairy, roots pretty easy in soil. Just keep it a little damp, not wet. Hope this helps. Nancy retsec

The one I showed above was bought at Harmony Foliage with Mr. Begonia (last year's ABS president) telling me the names as I was labeling everything. What can I say? I just go with the flow... That seems to be the problem with names in begonias (and everything else) - someone drops a tag and sticks it in another pot or you can't read someone's writing or everything is labeled rex begonia. If only someone would come out with a definitive book on (fill in the blanks here) Check out the international database of begonias for Muddy Waters. Ignore the 3rd photo since it is the one I just posted above. Begonia IDB of Muddy Waters

If you were going to neglect it then I would say filtered light or morning sun but if you check it every day or two for wilt, then several hours of full sun will be good for it - hardier leaves, more blooms, etc. The leaf color will change though but it is a toss-up - darker leaves or greener leaves and blooms? I let a big pot freeze a couple of years ago so all the stems were killed to the ground. I had a change of heart to save one of the other plants in it - a golden Duranta. By February or March, I saw some new shoots coming up from the ground and they put on the healthiest shoots/leaves I ever had. Unfortunately the golden Duranta didn't come back.