Pieces take root very easily. You can cut anywhere and stick in soil. It's safest to let the wound dry and seal at least overnight, but often I don't do that and haven't had any problems with rotting.
There are a bunch of these epiphytic cacti. If one is happy @ your house, it's likely so many of the others would too. This pot has about 6 different kinds, from various genera, some Epiphyllum and some others.
