This seems to be more of a boundary issue rather than needing to figure out the why of it. This answer is a collection of approaches that have worked for me in the past.
Redirecting cats with lightning rods
One of the main differences between cats and dogs is that a dog listens to your commands, but a cat acts based on what's in front of it. You cannot command your cat, but you can steer its behavior.
If your cat has a particular spot they like scratching, try putting the scratching post there. Whenever they go to that location, the scratching post sits there as a lightning rod for the cat's attention. If it likes scratching the scratching post more, then it will scratch the scratching post instead.
The further away you put the scratching post from the scratching spot, the less effective it becomes as a lightning rod.
This, combined with consistent negative feedback about scratching the other things, is usually enough to steer a cat's behavior in the direction you want it to.
Are you not entertained?
Your cat sounds like he's bored. Just like any creature, cats have a finite amount of energy. If you redirect their energy towards (allowed) playing, the naughty playing will lessen (or cease).
Play with your cat. Tire him out. We don't mind our cats running around, but it's a bit annoying when we go to bed. So we play with them just before bed to tire them out, so they're unlikely to want to run around the first few hours after playing with us.
He has a cat tower that includes a scratch post, and a scratch post that has hanging toys on it.
Not every cat likes every toy. Maybe your cat just doesn't like the toys he has.
Buy some varied toys and see which one they're willing to play with by themselves. For example, our cats love the laser, but they are only marginally interested in the automatic laser toy (it doesn't move as interestingly as when I do it). Some toys are a huge hit when I play with them, but a miss when the cat is left to play by itself. Some other toys, however, are a huge hit in terms of the cats playing by themselves.
This is a matter of trial and error.
How to train your cat
There is a lot of ground to cover here, and I can't list all of it. This is a more comprehensive answer of mine that delves into how to train your cat's behavior. Mostly chapter 4 is applicable, but I think the other points are applicable to your case as well.
Buying a plant sprayer can do wonders for training a cat, but you have to use it correctly. If you just use that as the immediate punishment without any verbal warning, you're not teaching your cat anything.
What works a lot better is a series of escalations:
- State the cat's name.
- Repeat the cat's name, but louder. Follow up with a clear "no".
- Take out the sprayer and restate the name/"no".
- Spray the cat. It doesn't need to be a direct hit, but it needs to come across as intending to hit.
- If the cat persists, calmly pick them up, bring them to another room, and put them down.
Always start from the top. If the cat does not stop what it's doing, take it to the next level. Whenever the cat yields, you reset the clock and start from the top the next time it happens, even if that is only minutes later.
The goal is to teach the cat a pattern of progression. Initially, it doesn't understand your verbal feedback. But after several experiences, it will start to figure out that these verbal cues are a precursor to a punishment that it really wants to avoid (= the spraying).
Eventually, your cat will learn that if it does not yield when it hears the verbal cue, it's goign to get sprayed. And since it doesn't want to be sprayed, it yields, and your training has succeeded.
Specific feedback
We have many tall windows that he likes to jump onto, but ends up scratching off the paint on our walls (we rent) when he jumps up there.
This particular instance seems to be a matter of unintended consequence. I wouldn't particularly fault the cat. Either find a way to shield the wall (we tape cardboard to the wall), teach the cat not to jump up there (more on how to train a cat later) or simply fill the window sill to a point where the cat can't be on it anymore.
Anytime I put my food/drinks or really anything on the table or counter, he goes up to it and tries to "clean up" as if in his litter box. This usually results in knives falling off counters, my drinks being spilled, his paws in my food, etc.
This burying behavior stems from cats trying to hide their remains of food because it hides their scent, and cats have a natural instinct to want to hide their presence in an area (which is also why they bury their droppings).
You can't teach the cat that this is wrong because their instinct compels them to do it. At best, you can train them to know that you will get upset with them for doing so.
He's currently trying to jump on our shelves that aren't very secure or big enough for him, so when he eventually does get up there, I know the shelves will break off the wall also causing huge holes and knocking heavier glass objects off that could hurt someone sitting under them.
This sounds like your house is not reasonably catproof. Cats like heights, you can't change that. If your shelves are fragile enough that a cat's weight will literally rip them out of the walls, and you put heavy glass objects on them, that's a recipe for disaster.
You can't reasonably expect your cat to fully tailor itself to what you'll allow and won't allow. Cats don't care about what you want, they care about what they want.
Oh yeah and he tries escaping out of the house any time we open the door. He has gotten out many times but thankfully we are able to catch him before he goes anywhere.
That's not really misbehavior. If you don't want your cat to go out, and it wants to go out, it's up to you to prevent it from doing so.
We try sleeping with our bedroom door closed so he can have the house, but he ends up scratching the paint and wood off our door and sometimes pulling carpet up.
Closing the door "so he can have the house" sounds like a bit of a misdirection. Even with the door open, they can still be in the house when they want to be. There's an ulterior reason why you're not wanting him in the bedroom.
Your cat clearly wants to be in the bedroom. You clearly don't want it there. Something's got to give. You can try and train your cat to stop trying to get in, but it's not going to be easy or a guaranteed success.
It may be a lot easier to address the issue which is leading you to not want the cat in the bedroom.
