Under what circumstances are one-way functions reducible to their hard-core bits?
Inverting a one way function is not always reducible to evaluating a hardcore bit. As an example, consider trivial hardcore predicates. Let $f$ be one way and define $g(x,b)=f(x)$ where $b\in\{0,1\}$. Obviously $b$ (the rightmost bit of a string) is a hardcore predicate for $g$, however knowing it does not bring you any closer to finding the rest of the string. The problem is that such predicates are probably of no use. Usually, we cant afford to lose information (e.g. while constructing pseudorandom generators), so we won't be able to base hardcore predicates on a part of the input that we're not using.

To overcome the above problem (i.e. ignoring trivial hardcore predicates), we can demand that evaluating the predicate will help us reverse the function (if this is the case, then $f(x)$ and $h(x)$ cannot be independent). The generic construction in Goldreich-Levin's theorem indeed yields such a predicate. If $f$ is one way and $g(x,r)=(f(x),r)$, then an algorithm computing $h(x,r)=\langle x,r\rangle$ can be used to find $x$ (just feed it $f(x)$ along with different values of $r$).