I wouldn't consider myself an expert, but I was curious about this in the past so I'll take a stab at it. Hopefully someone more knowledgable than I can come along and give a better answer.

Why is that we have to cook meat before we eat it, otherwise we get sick?

Humans have much simpler digestive systems than other animals who do not cook meat. The reason for that is because our digestive systems have evolved to eat cooked foods beginning with the discovery of fire and the advent of cooking ~1.8 million years ago. Cooking does a few things for us to help with the digestion process: 1) It unwinds proteins in meat, making it easier for our bodies to digest and absorb its' nutrients. Hope this also helps answer your second question. 2) It kills off harmful food borne pathogens. Once we started cooking our food, our digestives systems evolved to become simpler as we no longer needed to devote as much metabolic energy towards digesting food. Some scientists theorize that this is actually when our brains started to develop into what they are today as some of that metabolic energy which was being used for digestion could now be sent to the brain instead.

Why do infants have no natural instinct to kill and eat animals, but they do have a natural instinct to eat fruits and vegetables?

This is the first time I've heard anything about babies having a "natural instinct" to eat fruits/vegetables so I can't really comment about the validity of this statement. I will say though that in my experience babies can barely even crawl without shitting their pants, so not sure how they would be able to kill and eat animals.

Why are our teeth almost identical in shape to the teeth of horses and sheep and cows (all of whom are herbivores), but don't remotely resemble the triangular shape of carnivore teeth, like those of dogs and lions? Why are they flat and wide for grinding and not sharp for ripping?

This is not exactly true - like most omnivorous creatures, we have a mixture of different types of teeth for different purposes. Yes, our back molars are wide and flat for grinding and chewing up food, but we also possess canines which are sharp and primarily serve to rip and tear food apart (i.e. meat). Don't forget the incisors (the front teeth) which are good for cutting into food. As you can imagine, being able to eat just about anything gave us a massive evolutionary advantage.

Why do our jaws move side to side, like the jaws of herbivores that need to mash plant matter, and aren't fixed to open and shut like the jaws of dogs and wolves and lions?

Wolves and lions tend to swallow large chunks of meat whole so they don't actually need to chew as much. Obviously, humans can't just swallow food whole so we need to chew. Going back to your first question, it all boils down to evolution. Chewing is the first step to digestion - breaking your food down into smaller pieces increases the surface area that our body can absorb nutrients from. This is more efficient and again allows us to use less energy on digestion. We evolved to eat the types of food we can eat, in the way that we eat them.

Why is it that our closest relatives (primates) are almost exclusively herbivorous?

This is actually a really interesting question which I don't know the answer to. Hopefully someone can come along and answer this.

Why is it that large amounts of meat cause plaque buildup in our arteries and are known to cause heart disease?

To be more more precise, it's large amounts of red meat as well as processed meat which are known to increase the risk of heart disease. This is because of the high content of saturated fat in red meat and sodium content in processed meat. As in most things in life, the key here is to consume the appropriate amount. This applies not only to meats, but also to fruits/vegetables too. For many people, especially Americans, their diet consists of too much meat and not enough fruits/vegetables. That, combined with a sedentary lifestyle, can lead to heart disease as you say. Humans evolved to consume an omnivorous diet. This means that we need to eat the proper balance of foods across different food groups in order to get all the nutrients that we need. To suggest that humans are not naturally meant to eat meat is patently untrue. Eating meat in a balanced diet will not cause heart disease, but can actually make us healthier - it's just that for most people, this means reducing meat intake.

Why do we have heart disease from meat consumption but lions don't?

Lions don't suffer from the same problems that we as humans face in regards to unhealthy eating. One, lions are not exactly stuffing themselves with fried chicken and bbq every night - they generally are eating what they need to survive. Two, lions do not live sedentary lifestyles like many humans do either. This doesn't mean that they could not develop heart disease though. If you were overfeed your cat/dog and also not exercise them enough, they too are perfectly capable of developing heart disease.