Is it generally true that for random variables $X$ and $Y$, regardless of being dependent or independent, that $E[X] = E[X \mid Y \le a] + E[X \mid Y>a]$ ?
Not quite, if we use the law of total expectation we would have that
$$ E(X) = E(X| Y \le a)P(Y \le a) + E(X|Y > a) P(Y>a) $$
Exactly Chappers, and I can just add if $X$ and $Y$ are independent continuous random variables, $E[X|Y]=E[X]$.
So Mauro, your formula would be false because in this case, you'll have
Finally, regardless the dependence between $X$ and $Y$, we have always: