Your pics:

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General rule is to only pot up as the plant fills the pot it is in. If you want a big plant then I find a big pot is best but I'd do it when you can move it outdoors. Come fall you would probably want to "pot down".

Regular feeding is recommended but a lot of potting soil has slow release already added to the mix and is good for many months.

Trim as you see fit. Grown in a big pot with the best light, regular feeding, warm temps and your begonia can reach 4 feet or more. Or you can start it over once a year and keep it on the short side. Kept in a small pot will constrain the roots and growth (bonsai in a way).

Cut the tops out and start them in a new pot or add to the pot it is in. The stumps will send up new shoots and if not cut to the ground they will grow new shoots at the growing nodes. The stems in between can be used to start new plants as well and they can be placed horizontally in a tray with some soil or vertically (make sure the bottom cut is the one that goes in the soil though).

Your begonia looks like Lucerna - an oldie but goodie. Very easy plant to grow and keep for a long, long time. The only problem I've seen is mildew in the fall/winter in a basement.

Hcmc, I thought it looked like Lucerna/Lucerne also. With more light, the leaves can develop a nice reddish tint if that's what it is.