An intelligent adult dog and a young child work out to about the same intelligence level and from experience taking a dog to the vet can require some subterfuge, they don't seem to understand sentences but they can pick up what you're talking about by indervidual words and your overall tone. I once tried to interview a dog, a German/Belgium Shepard cross, a small bear of a dog, wonderful temperament, so I sat down with him and explained what I was doing and asked if he understood and if he did he should bark.
He didn't bark, but by the way he looked at me I could tell he knew I was talking to him, he knew I wanted him to do something, listening intently, studying my eyes, it's rare for a human to speak to a dog like this and he was unusually focused, but he didn't bark, he only reacted to key words and tone, so I think he might have been intelligent enough to understand me but a dog's brain just isn't wired that way, it would be like if someone was speaking to me in Japanese, I'm not fluent in the language but from key words and facial expressions I can deduce the gist of what's going on.
Do animals know we're more "intelligent/advanced"?
I don't think they think about it that way, there's this Australian newspaper comic called Footrot Flats and in it there's "The Dog" our protagonist and Wal who is his owner, at one point Wal is out in a paddock, catches a ewe and has to tie her up so he can go get his truck and take her off to market, but he's out in the middle of the paddock and has nothing on hand to tie her up with, a predicament the dog realises and finds amusing. So to Dog's surprise he takes off his belt and ties her up with that and as he ambles off to get the truck Dog follows along, mightily impressed by Wal's apparent foresight, while Wal is struggling to walk and keep his pants up.
I don't think cats & dogs think they're smarter than they are because I doubt they have a conceptual understanding of intelligence, when a dog sees a man fixing a fence he doesn't know why he's doing it, the dog doesn't know what the fence is for, but the dog doesn't care, to a dog a fence is just scenery, at most an obstacle.
But if a man is carrying fence fixing tools a dog will understand that he's a out to go fix a fence, it may even know where he's going if it knows of a section of fence that's obviously broken, but it won't guard that broken section to prevent animals getting through, or if it does you've got an excellent dog and you should make damn sure it breeds.
Cat's on the other hand, I used to live with a jet black Persian called Ebony and she was wonderful, if we were all sitting out on the deck playing UNO she would sit in an empty chair and we would take turns playing her hand. Of course she didn't understand what was going on but it was social and she wanted to be a part of it, not as a pet but as a person, mind you she was raised around people and rarely if ever interacted with other cats so she might have thought this was perfectly normal.
