Running multiple licenses on the same Battlenet at the same time can be tricky - and there are limitations.
Normally, we suggest you setup another email (Battlenet) under your credentials for a child until they come of age where it can be in theirs.
To the first question, you should be able to exit the launcher after the game is in progress, then use it on the other.
Some of the limitations - chat and pet battles can go to the other account, depending on which one is logged in first.