Here are some comments on the questions in the problem (but not full solutions):
What is the probability of H_0 being rejected?
Do your answers change if the true proportion of greens in the bag changes?
What would happen if you changed the hypothesised proportion \pi?
What would happen if you changed the significance level of the test from 5% to 10% or 1%?
This depends on the proportion in H_0, the true proportion, the number of trials and the significance level. We can get evidence from the simulation, or we can work theoretically. In general, we would expect that the greater the difference between \pi in H_0 and the true proportion, the greater the probability of H_0 being rejected (the null hypothesis is "more wrong"); the
greater the number of trials, the greater the probability of rejection (the sample proportion will be more likely to be close to the true proportion), and as the significance level is raised, the probability of H_0 being rejected will also increase (as we are reducing the range of acceptance).
The probability of rejecting H_0 in this problem can be calculated as follows. Let the hypothesised proportion be \pi_0 and the true proportion be \pi_1. Let X be the number of greens observed after n trials. Under the null hypothesis with significance level \alpha (so typically \alpha=0.05), X\sim \mathrm{B}(n,\pi_0), and the null hypothesis will be rejected if
X lies in the critical region, which is Xx_2, where x_1 is the largest integer for which \mathrm{P}(Xx_2|H_0)\le \alpha/2. We can then calculate these probabilities given that H_1 is true, so that X\sim \mathrm{B}(n,\pi_1) and deduce that the probability of H_0 being rejected is \mathrm{P}(Xx_2|H_1). These calculations can be easily performed by computer.
Note that it is only possible to perform this calculation if we know the actual proportion. But if we know the actual proportion, why are we doing a hypothesis test?! This makes the power of a test a somewhat difficult idea. We could, though, be more specific, and say that we are testing H_0\colon \pi=0.5 against H_1\colon \pi=0.6, and
ask which of these hypotheses is more likely to be true. This is a different way of performing hypothesis testing, which is dealt with in the article [yet to be written].
If H_0 is rejected, how likely is it that the alternative hypothesis H_1 is true?
A tree diagram will help here: we have two possibilities, H_0 is true and H_1 is true. And for each of these, either H_0 will be accepted or rejected. So we have, looking at the tree diagram [which would be nice to draw]
But we don't know the majority of probabilities in this calculation! We only know that \mathrm{P}(\text{$H_0$ rejected} | \text{$H_0$ true}) is the significance of the test, which we have chosen. So without some idea of how likely it is that H_1 is true, and some idea of the probability of rejecting H_0 if H_1 is true, we cannot say how likely it is that H_1
is true even if we rejectH_0! Likewise, we cannot say how likely it is that H_0 is true if we accept it.
If Robin wants to be 90% certain of rejecting the null hypothesis if it is wrong, how many trials are needed?
This again depends on the actual proportion of green balls. If, though, Robin assumes what the actual proportion might be, we can then use the above calculations, trying different values of n until we find one that is large enough so that \mathrm{P}(Xx_2|H_1)>0.9.
Remembering that each trial costs a certain amount, what is the best number of trials to perform? (And what does "best" mean?)
This is a hard question! It depends on what is most important to Robin. It is a balance between getting the "correct" answer, avoiding the "wrong" answer, the cost of the trials, and the assumed alternative hypothesis actual proportion.