Hi, Skywriter and London, welcome to the forum. Now, let me clarify a few points. ALL parrots dump their food and water bowls but cockatoos do it more than other species. This is a natural behavior, part of their ecological niche (they disperse seed and feed ground species) so 'fighting' it is useless, what you have to do is get the kind the locks in place like this one:
https://www.amazon.com/Lixit-30-0740-01 ... 60&sr=8-23 You can't visually sex a cockatoo, you need to have it DNA'd - this is actually easy to do and inexpensive (see this:
https://iqbirdtesting.com/)
ALWAYS allow the bird to come out of its cage before you clean it because pet parrots that are not kept at a strict solar schedule and on a good diet get overly hormonal (cockatoos are considered one of the most hormonal species of parrots because they have two breeding seasons a year) and that translates into aggression and cage-possessiveness).
Please stop feeding him a Walmart parrot mix immediately - there is NO good parrot food in Walmart (at least, there isn't any in all the Walmarts to which I have been). Cockatoos eat A LOT of raw produce and should never be free-fed any type of protein food (seeds, nuts, pellets, avicakes, nutriberries, etc). The right diet for a cockatoo is gloop and raw produce for breakfast (one large piece of fruit, one large piece of veggie and one leafy green or a cruciform - a different one of each every day of the week if not even more seldom) and nuts for dinner. This, with a good quality multivitamin/mineral supplement given three times a week (this is only if the bird has been getting vitamins all along, if not, you need to give it to him daily for two to three weeks to replenish the deficiencies). No peanut butter (it's full of salt and sugar, two things parrots should never consume as added products) and, although bananas and apples are good, do try to give it blue produce (blueberries, blackberries, concord grapes, black currants) at least twice a week - and be careful with kale, it has a huge amount of sorbitol, an indigestible sugar that can cause diarrhea - try raw broccoli, instead (broccoli is the king of greens, sweet potatoes are the kings of the veggies and blueberries are the kings of the fruits).
Now, parrots take a couple of months to start to feel comfortable in their new homes so be patient with him. The best way to get the used to the new home fast is to keep the schedules unchanging so the bird can foresee what is going to happen and feel safe. The day starts at dawn (around 5:30 am this time of the year in USA) when the cage should be open so the bird can come out, the cage cleaned and the breakfast and fresh water served (I find it best to feed the birds that live in cages inside the cage). One breakfast is done, the bird can come out again and this is when any type of interaction is best because this is the time of the day that the wild birds interact with one another so things like a nice spray bath, a game or some sort of exercise or entertainment are all good.
BUT, the MOST IMPORTANT thing with a cockatoo is to always, always, ALWAYS, have plenty of chewing material for him - in my personal experience, they like nothing better than a big, hard carboard box (big enough so they can get inside) but a 2X4 cut into cubes, natural branches, even wood planks as long as it's untreated are good - they just need to chew, chew, chew. Oh, and music! They really enjoy anything with a strong beat and, if you dance for him, he will dance with you!
Don't worry too much about the cat. I have a lot of cats, a lot of dogs and a lot of parrots and they all cohabit without a problem - the trick is constant supervision (never leave the bird out of the cage and alone with the cat and, when they are both in the same room,, watch them) and, if the cat shows too much interest, a good cold water spray on his face along with a command (I actually use sounds with my cats and for them to leave something alone, I give them a good strong loud SHHH! and they take off) will teach him not to stalk the bird. You might also want to consider feeding the cat high protein food if you don't already do it. That's what I do. I started doing it because it's healthier for them (I am a maniac when it comes to what my animals eat) but I read a study recently that found that cats that eat high protein (real meat as first ingredient and not less than 35% - I feed 40% to my cats, all meat and no fillers because it was found that less than that will result in muscle mass loss over time - my cats usually live into their twenties) do not hunt as much so get your cat some good, high protein food, it will help.
Let me know if I covered all your questions and feel free to keep on asking.