Should the keys for pets_allowed and dog apply to just the outdoor seating area, or the operator (restaurant too)?

Interpreting freely, =yes would mean a yes anywhere, =no a no everywhere. You could try outdoor_seating=yes + outdoor_seating:dog=yes + indoor_seating:dog=yes but this has no support, only informative use to editors and future reference. It would be best to add dog=yes to a leisure=outdoor_seating, and dog=no to eg the building=retail. This doesn't replace the dog=yes on the amenity=restaurant.