Sorry to hear about Ling-Ling, I'm sure she will be fine. The lumps are offsets, a.k.a babies. Cacti like that never grow singly, they always make lots of offsets until they form a big clump. I don't think that the falling damage is serious, just remember that a larger pot will take longer to dry out and be careful not to overwater and cause rot.
Likewise, Cacnea also likes to grow in clumps and that's why there's a new offset and it is normal for the stalk attaching them to the parent plant to be very thin. Just remember that they need to dry out for a week or two after you cut them and be careful not to overwater while they are rooting. It takes a long time.
I must say that all of them look like they need a little more direct sunlight if possible