Help on dog training?

If a dog seems to understand the meaning of NO when you say it to them when they do something wrong, why would they do these “wrong” things when you’re not around? And how does one stop them from behaving this way when you’re not around?
Example, i’ve told my puppy many times NO when she goes near my cat’s food, she’s not allowed to eat it, so i’ve moved the kitty’s food onto the couch, but now puppy has discovered she can climb onto the couch (she’s been too small to do so previously), and now she eats the cat’s food, even after i’ve reprimanded her for eating the food on the couch one time). But when i’m not in the room, she’ll go back and do it.
And, she knows to stay out of my crafts room, because i can’t have her chewing on my crafty stuff as i often work on the floor because i need the space. Yet sometimes when i’m not around, she’d follow my kitty into the room or just walk in on her own, but when i’m in the room or sometimes not even in the room (i spy on her from down the hall, haha) she’d sit outside knowing that she’s not allowed in.
Any suggestions? Because it’s driving me nuts and testing my patience because i don’t know what i’m supposed to do.