At 8 months, I am surprised that you are still gating him when you are home. I understand that you need him out of the way when you're mopping the floor or something and you may not want him loose when you are taking a shower, but both of these things should not take more than an hour or so. Why not keep him with you when you can and crate him when you absolutely cannot have him underfoot or on his own?
The dog has already learned that he can jump the gate and when he does, he is rewarded (he gets to be with you). For many dogs, that is worth the "punishment". Apparently, he just wants to be able to hang out with you. Why not allow that? He is a young Lab, afterall. That's what they do. Teach him the place command. Use bones and filled Kongs frozen with treats (or whatever works for him) to keep him occupied. is it the cat? Can you teach him manners around the cat? Allow the cat sufficient places to get away from the dog?
He also does sound bored and yard time and walks is not enough for a young Lab. Is he a field bred Lab? What about swimming? Retrieving? Running off leash? Playing with other dogs?