I have seen many a dog raised by cats- including in our own household. Ours mimicked the grooming behaviors and definitely deferred leadership to the boss cat.
The funniest of them all, however, was a friend’s red Dobie (60+ lbs) raised by a pair of smallish cats (both under 10 lbs). Not only did the dog swat preferentially over biting, he was fully part of cat-style zoomies.
IOW, as we’re sitting on the couch playing a video game or watching sports or monster movies, if you heard zoomies begin, you needed to be on alert. If the couch was going to be run across while you were on it, the 3 of them could all run over you, in any order, in any direction. And often, more than once.
For example, a zoomie sequence might be:
DOG => cat => cat
cat <= DOG <= cat
DOG <= cat <= cat
cat => cat => DOG
And so on. That dog would absolutely destroy your lap & legs and…everything.