Is there a problem with eating eggs uncooked?

Indeed there is - you could easily get Salmonella which is common in eggs but the bugs can be killed off if they are cooked properly

Salmonella

https://en.wikipedia.org/wiki/Salmonella

You can get sick from bacteria like Salmanella and can make you very sick...there's other ways to get protien, like cooked eggs

Yes, chickens tend to frown on it and they gather in a flock and taunt you

You could get salmonella.