Thank you for your question. You have a few different issues within your one question, but let's start with your main question regarding cats eating each other's food.
I have the same problem with my cats. I have one cat who gorges herself, and I have one skinny cat who likes to graze all day long. What do you do with this situation? It is not easy. You have to get a little creative. And unfortunately you will likely never get to the point where they will only eat their own food. However, you can get to a point where they mostly eat their own food.
One thing to note is that cats tend to be dominant over certain things. Cats are not usually dominant over everything, like dogs tend to be. Some cats are dominant over food, which it sounds like your cat is. My thin cat will not eat unless my overweight cat is eating. You can see the obvious dominance there. Here are some ideas for you...
KEEPING CATS FROM EATING EACH OTHER'S FOOD
I feed my cats wet food at night. I feed the slightly overweight cat in one place and my thin cat on a high shelf where she can see the other cat eating.
During the day, I feed my overweight cat using an interactive feeder, so she has to use her natural instincts to work for her food.
I give my skinny cat a small bowl of dry food on another shelf, so she can graze. It's not that my overweight cat can't get to the other food, but she is content to eat her easier-to-get food.
Each morning, I also feed my skinny cat dry food in our closet, which is at the end of our walk-in bathroom. So there is quite a buffer from the bathroom door. I turn on the fan, so the other cat (who is closed out) cannot hear my skinny cat crunching. The things we do for our cats! I also sneak my skinny cat treats now and then.
CATS EATING DOG FOOD
This is a real problem for inside cats. You have to stop this. Even though your cats may be healthy now, eating dog food will make any cat overweight and unhealthy. I have volunteered at an animal shelter for many years and have seen this problem over and over again with overweight cats given to the shelter. Feed your dog at times you can supervise and take up the bowl when your dog is finished. If you kennel your dog, feed him in his kennel, so the cats can't get to the food.
CAN CATS EAT VEGAN FOOD
This is the most difficult part of your question because I can tell you have strong convictions in this area. However, I must tell you that cats should not ever be on a vegan diet. I am very sorry. I know this is difficult to hear. Cats were built to be obligate carnivores. They must have the taurine from the meat because they cannot produce it on their own, like dogs can. I know some vegan pet foods have synthetic taurine, but I wouldn't chance this. It is just not the same. Cats on a vegan diet will likely have severe medical conditions. See the quote below from Pet WebMD
If allowed to continue long enough, these dietary problems can lead to serious and sometimes irreversible medical conditions. The one veterinarians mention most often is taurine-related dilated cardiomyopathy (enlarged heart with weak contractions and poor pumping ability). Low taurine can also lead to reproductive failures, growth failures, and eye problems.
I hope this helps and at least gives you some ideas. I wish you and your animal family the best!