How old does a child need to be in order to stop sharing a room with their mom?

Mostly that's going to depend on what sex the child is and what living arrangements there are at mom's house. Most but not all states don't really regulate down to that level. Its more a matter personal preferences.

Yes, there will be "rules" if there are home inspection from Child services or some abuse is reported. But in general there is no "law" It is often a issue only after something has happened or is reported