Same way you do people, smash food up to a point where they can eat it
I'd say some kind of soft food. I'd ask a vet which is best for your cat.
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the best foods for your teeth are
Dry cat food. Plus it is also good for them and their teeth.
Fresh, raw, free-range meat is the best for all cats. If you are looking for cat food, I would suggest wellness cat food or EVO cat food, which are some of the best. Friskies, Fancy Feast and Whiskas should be avoided, as they contain bad things such as grain and meat-by-products. As a side note (though I'm getting off topic here), dry cat food does not clean teeth, raw meat is the best for that.
No, nor does eating dog biscuits whiten your teeth.
Yes a cat can cuz they are smart
Usually, absolutely not you shouldn't have your cat's teeth removed. If there's a medical condition with them, you should see the vet about it. If your cat didn't/doesn't have teeth, well, yeah, you're going to have to buy them soft food because obviously they can't chew the dry food without teeth.
Your cat should be mostly fine and able to take care of itself. Maybe switch to wet food if you haven't already. Dry food puts unnecessary strain on the remaining teeth.
Ideally, no. A cat should not eat/drink soup. Human soup is full of salt, garlic and onion which can be toxic to a cat in large amounts. Soups do not have all the nutrition a cat needs, as they have evolved to get all their nutrients from meat, and have sharp teeth to tackle raw flesh and bone. If your cat does not have teeth, the best food for it would be mashed up wet food so the cat can still eat without needing to chew.
The best cat food really depends on your cat. In theory you would want to feed your cat a food that is made with real meat, and no red dyes, as red dyes aren't good for cats to digest.
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Molars.
Cats eat with their mouths. As does every other mammal.