How can one prove that:
$$ E( E(Y \mid X) \cdot X)=E(Y \cdot X)$$ if $E(Y \mid X)$ is well-defined.
Are we free to use the law of iterated expectations? I am confused since now the expectation of the product is not the product of expectations and I am not sure if we can bluntly apply lie and just replace the conditional expectation with the random variable itself.
The key ingredient to the proof is that $\mathbb E[X Y \mid X] = \mathbb E [Y \mid X] \cdot X$; once you have established this, then you can immediately use the law of iterated expectations. (How you prove that lemma will depend on whether you're doing measure-theoretic probability or not, but it's not too bad in either case.)