Can you run water service on exterior wall?

Yes you can... but I wouldn't. In areas where there is a chance of freezing, (I live where the temps can get below freezing for up to a month at a time)it is against better judgment to put any pipe outside of the thermal envelope. In areas where Freezing is of no concern, pipes can be placed in any wall. There are Precautions that can be taken when working in freeze zones, such as heat tape, thermal wrap, etc. but I would stick to bringing the pipes through the floor and into the botom of the sink cabinet. this is the most common route in freeze zones, keeping the pipes within the thermal envelope. Hope this helps. Terry