I have no experience of over-wintering plants so I can't comment on the transition from basement to outdoors that you have experienced.
However having grown BOP both in pots and in-ground, I can say that they can achieve a very substantial root system, and I think what you've described are roots. And if so, I don't believe the broken part will re-generate.
The most usual way to multiply these plants is either by seed (slow growth) or division of the crown/clump but that is a bit risky as you may lose all of it.
I hope your plant survives, and without stressing it too much I'd suggest a replant to a larger container might help it re-constitute. Don't overwater as they are a very hardy plant and will probably fight to survive.

They're not actually a tropical plant. They come from a mediterranean climate, hot dry summers and wetter cooler winters. So they like a lot of sun. But like any plant, moving from complete shade into direct sun needs to be done gradually.