I have an XBOX 360 in my living room, and one in my kids room (on the same router). In order for bot units to be on XBOX live, do I need a family account, or just 1 standard account? Can both boxes be logged in as the same user at the same time?
No, only one Copy of the Profile can be Signed into Xbox LIVE at any one time from any one Console. I'm not quite sure what you mean "bot units"? As for a Family Plan or not, that's up to you if you have more then one User who
is going to be using Xbox LIVE Gold.
You'll need one account for each person that expects to have their stats saved for each game. My wife & I play on one Xbox (because the old elite RROD out of warranty) so each of us needed separate accounts in order to play games & save stats.