The instrument in "The Colonial Origins of Comparative Development" I am hoping to understand when an instrument is appropriate. For the purposes of this question, I'm considering a simplied version of the model in AJR (2000):
$$\begin{aligned}\text{GDP} &amp;\sim \text{Exprop} + \text{Latitude} \\
\text{Exprop} &amp;\sim \log{\text{Mortality}} + \text{Latitude}\end{aligned}$$
My questions are:

Does latitude need to be included? (If we believe that latitude has a causal effect on settler mortality, expropriation, and GDP, then my guess would be yes, since leaving it out would leave a correlation between log mortality and the residuals of GDP, through the effect of latitude?). Similarly/in general, would we need to control for anything that could possibly affect both the instrument and the outcome?
Do we need to assume that the entire causal effect of log mortality on GDP is mediated through expropriation? Would it not be an appropriate instrument if the log mortality -&gt; GDP causality could be mediated through something else (e.g. genetic diversity, folk traditions not related to institutions, etc)?
You should include all controls in the second stage, even if they are not instruments.
The entire effect of log(mortality) should be mediated through Exprop.  This does not mean there cannot be other things causing GDP or Exprop, but log mortality cannot have a effect on GDP except through Exprop.