Is the equation "$Y=\mathbb{E}[Y|X] + error$" an identity? Can I always use this equation to regress $Y$ on $X$, if I know the distribution of $Y$ to get an expression for the expectation term?
 A: Yes, it is an identity - just define $error:=Y-E[Y|X]$. 
Whether or not that identity is helpful in the regression you mention is another matter.
For one thing, the conditional expectation may be a nonlinear function of $X$ (may for example include squares of $X$), so that a standard linear regression may not estimate this function consistently (nonparametric approaches may be preferable).
Another problem is whether the conditional expectation, even if it were linear, corresponds to the object you are interested in estimating.
For example, a regression of earnings on years of schooling can estimate the expected earnings of an employee with a given level of education. But: the additional earnings related to an additional year of earnings will generally not be the same thing as the causal effect of such additional schooling, unless you manage to control for all potential confounders (experience, ability, and many more) in your regression (or have another identification strategy, like a convincing instrumental variable).
