It won't hurt to bring it with you. You can keep it on pause when you're busy, and unpause it when you have nothing else to do. Pausing it won't affect it in any way though, so it's perfectly safe to do.
Dont leave it paused for a week unlese you dont mind stalling its growth. You see pausing slows down tama's growth progress. If you dont mind, you can leave him paused but he wont change quickly.
Since putting it on pause uses up battery, you can always just take out the battery or put a piece of paper in between the battery and the metal piece underneath the battery (like tabbing it, how it comes in the package), that way no battery is wasted, and when you put the battery back in, you can left off right where you started.