Does absence of endogeneity problem means absence of heteroscedasticity?
For example, if an estimator is designed to correct for endogeneity does this mean that heteroscedaticity will be corrected for, too?
No, not at all. Endogeneity is a first-moment problem, while heteroskedasticity is a second-moment problem.
For example, consider a linear model
$$ y_i=x_ibeta+u_i $$ An assumption like $$E(u_i|x_i)=0$$ implies no correlation of error term and regressor, so no endogeneity. No (conditional) heteroskedasticity in turn can be written as $$Var(u_i|x_i)=sigma^2,$$ where $sigma^2$ is a constant number.
There is no reason whatsoever that $$E(u_i|x_i)=0$$ would imply $$Var(u_i|x_i)=sigma^2.$$
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