What happens if you eat partially raw chicken?

It depends on whether it's infected. Chicken is very prone to infections such as salmonella and that's why the advice is to cook it until it's white ALL the way through. But if this meat came from a nice healthy uninfected chicken, you'll be fine.

Chicken itself is never the problem - it's the bugs that can get into it. Pork is similar so that's another meat that should be thoroughly cooked. Which is the original practical reason why Judaism and Islam forbid anything that comes from pigs. In hot countries centuries ago with no refrigeration, it was just safest to avoid it. Beef and lamb, on the other hand, are much safer and there is nothing wrong with a nice steak that is still pink inside. (It's nicer that way. My mother always used to cook roast beef until it was brown all the way through and slightly burnt on the outside. It was horrible - like I would imagine cooked shoe leather would be like. Of course I was a good boy and dutifully chewed my way through it but it really was totally inedible without some gravy.)

Nothing. The FDA puts out these scare bulletins to keep their budget intact every year.

People have survived thousands of years eating undercooked flesh. You'll be fine.

You can get salmonella and end up very sick.