CIA and Exclusion restriction
The CIA assumption is equivalent to $E\{ \nu_i D_i \vert X_i \} = 0$. However, it is not the case that the unconditional expectation is equal to zero. The whole reason we need the CIA is that we don't have random assignment, but potentially we have random assignment conditional on the covariates we can observe. 

I think it may help to think of $\nu_i$ as an error term that is composed of other variables $Z_i$ with effect $\gamma Z_i$. In this case, $\nu_i \perp D_i \vert X_i$ means that $Z_i$ does not predict treatment after conditioning on the observed covariates $X_i$. 

To answer your IV question, this is indeed similar to the exclusion restriction in that you are making assumptions about the unobserved variables $Z_i$ in your data generating process and your treatment/instrumental variable. But I wouldn't try to connect the ideas too much or else you may come to wrong conclusions.