Does a (properly) randomized experiment eliminate all common sources of endogeneity problems? By common sources of endogeneity, I mean: measurement error, simultaneity, and omitted variables (Wooldridge 2002). I am fairly certain such a experimental design eliminates the omitted variables problem - but what about measurement error and simultaneity?
No, it doesn't.
When feasible, it is an important way to deal with the aspect of omitted variables that leads to bias (though not the impact on error variance by the incorporation of omitted variables into the error term). It doesn't do magical things like get rid of measurement error (or, as Repmat adds, does it deal with simultaneity; that's another one of those things it would need to be magical to deal with).
The answer depend of what you mean with "properly". In an idealized (properly?) experiment we are sure that problems as omitted variables, simultaneity or measurement error or also any others source of endogeneity don't play any role. If the experiment is idealized any of these problems way out by definition. Also in terms of measurement errors, in idealized experiment, we don't have any problem to measure the effect of the tratements on the subjects. Probably you are referred to econometric problems and in that field, tipically observational, if we want to capture the average causal effects we would to achive a scheme that is as good as an experiment. However in the real world no experiment is perfect (idealized), at the best is close, and some problems can occour.