i) Correct; I would have accepted any reasonably justified answer around 0.5% (as there are several ways to interpret 'false positive'). 2 marks, good answer.
ii) Incorrect: 0 marks.
The reason a high number of the positive test results were false positives was because, as stated in the paper, the study was completed when the prevalence was very low (June-July). Consequently, the number of true positive tests will be very low, while the fraction of false positive tests remains high. Thus when the prevalence is very low, a significant fraction of positive tests will actually be false positives. This is well-known and understood.
iii) This question could be fairly disputed as it's arguably a _little_ bit unfair. As the prevalence of the virus has grown dramatically, the fraction of false positives will have remained the same but the fraction of true positives will have increased dramatically. Therefore the fraction of positive tests that are false positives will go down significantly - instead of most positive results being 'false', most positive results will be 'true'. However, the fraction of false positive tests will remain the same, so the same fraction (<0.5%) of tests will result in the wrong advice (to isolate, to miss work) or the wrong treatment (moving into a Covid ward, for example). So arguably while the tests become 'better' at higher prevalence (more likely to be correct), the problem hasn't really gone away.
So, in your infinite wisdom: what do you suggest doing about this very small fraction of tests that results in poor treatment? Doctors will (or should) know about the risk of false positives; I suspect most positive tests will, where possible, we followed up by a second test to verify.
The false positive rate for the ONS study was less than 0.1%, because the false positive rate cannot exceed the minimum positive rate (once accounting for a bit of statistical uncertainty).
Fair play for answering, though...