Let X be the number of people achieving the stated IQ. We are given that and clearly . First, you need to solve for using the information given (you know the moments, so you can recover these parameters).
Getting the prior estimate is trivial; it's just .74. The MLE for theta in 2007 is also trivial; just the sample proportion. To get the posterior mean you need the posterior distribution. Use the fact that the posterior is proportional (treating X as a constant) to the joint distribution to get that. It happens to be the case that the posterior is , but you should make sure you can do it working with the densities. Answering the last question just requires knowing the mean of that.