Cats and litter: Cats are generally pretty good at being litter trained, as by nature they want to hide their waste AND older cats will usually teach younger cats how to do it. Kittens once trained are still messy and have accidents - though pooping in a corner is a sign it is deliberate. Reasons a cat may deliberately go outside of the litter could be: the litter was too dirty, the litter was occupied, the litter was unreachable, the corner they went in was dirty enough for them to think it was a litter area and lastly stress.
They may just be learning/getting used to their environment. I wouldn't be tooo worried yet but the general advice is one litter try per cat plus one extra. You can downsize this when they're older and more in a routine, or just get a really large tray and dump a whole 10kg of litter in it (what I do for my maine coons).
Teaching them where they can't go: generally, spray bottles do not work. They just instil fear of water, spraying noises and bottles in to them more than they do stop the negative behaviour. Cats aren't like dogs. It's difficult to attach the negative enforcement with the fact they were on the table, water spray can often teach a cat to do bad things just when you're not around, they're crafty like that. Deterring a cat works better, i.e. giving them spaces they can climb and luring them away with toys and tricks.
If you find that isn't working and some people may disagree with this... A simple flick/boop on the nose is better than a water spray. Yes, it's sad you have to resort to something physical - but it doesn't have to be hard. If you accompany it with some kind of keyword for bad (mine is just "OI!"), you can eventually just shout "OI" and they understand what you mean.
Stomach pets: Most cats hate it. Simple as. You know the 'death kick' or 'rabbit kick' cats do (where they grab something then go nuts with their hind legs)? This is because instinct is trying to disembowel their prey. Cats understand that their bellys are soft, unprotected and vital to their survival. The best thing to do is just wait until they present their belly to you and be gentle at first. If they never present the belly, there is no way to force it sadly.
My cats both loved belly rubs from when I got them, so I can really go rough on them similar to a dog and they love it - but others, that'd be a death sentence. Cats have very unique personalities that are hard to change. You just need to learn what kinda personality your cat has.
Leashes and stuff are best started early. Cats are best trained when hungry so you can skip a meal time to then work on training. Things like just getting them used to the leash (playing with it, touching it, etc) should be rewarded with food and then you can slowly work up to getting them used to wearing it.
Thanks for this break down - I think the current litter box is about 12 inches by 15 inches but I've seen both sit in it at once. I'm not opposed to a second litter box, just dreading the thought of weekly cleaning of two boxes. Those are some really good tips for training, I'm really trying to discourage them from jumping on the table when I'm on the laptop.