Relationship between covariance and conditional expectation

I read in my introductory stats book that if we have 2 random variables, namely, X and Y, then if Cov(X,Y) = 0 then this implies E(Y|X) = 0 likewise if E(Y|X) = 0 then this also implies Cov(X,Y) = 0. How do you prove this? It doesn't seem very obvious to me...

Also what is the formal difference (I guess I'm looking for the formal definitions) between an estimate and an estimator. I have an intuitive idea but I am not sure how to formally define it.

Thanks