Specifically regarding ancient Greece and Rome, how common was it for statues to appear much better looking than who they portrayed? It is forbidden to ask for examples so just some general background or something to guide me the right way in examples of this if there are any.
Fairly common, but not for the reasons you might think. It wasn't just that individuals wanted to be portrayed as better looking than they really were.
Today, we tend to think of portraiture art as being representative of a person's physical characteristics. If we're painting a portrait of a man with a big nose and red hair, we'll probably depict him with a big nose and red hair. Yes, we might indulge in some artistic license; we might make his nose a little less bulbous, his hairline more full, so that he'll be happier with the portrait, but generally speaking we will try and depict him fairly accurately.
Ancient art was often concerned with representing different characteristics, primarily personality traits and accomplishments. In Archaic Greece, the majority of human statues are depicted in a fairly typified manner, without much attention paid to individual features. This is another example of the different concerns of ancient artists, which has been attributed to the idea that the notion of the self is fairly recent, and that it was important to depict people based on what, rather than who, they were. That sounds a bit odd, but essentially there is a school of thought that the idea of the individual as being more important than the role they played within society is not actually a universally meaningful one. This meant that statues of human figures, even tomb art, tended to be of a type - e.g. nude men standing, clothed women standing - rather than of specifically recognisable individuals.
In Classical Greece, individual representation was still limited in favour of typified sculpture, but this was often achieved through the use of idealised body types. A man who was considered to be heroic or influential, therefore - or indeed a man who wanted to present himself as heroic or influential! - might commission a statue of himself depicted in ways which emphasised these traits, rather than any of his accurate physicality. To extend the example, if we were to sculpt that same man as previously mentioned but in Ancient Greek style, we would look more closely at his other attributes. What was his status within society? Was he a politician, a philosopher, or a religious leader of some kind? Was he liked or disliked? And importantly, was he commissioning the statue? All of these questions would go into deciding how he should be portrayed. Specific Classical ideas of beauty, best witnessed in the plethora of male nude sculptures, were popularly used.
Although of course there are some exemptions to that rule, often regarding funerary and grave art, it wasn't really until Hellenistic Greece that art depicting individuals more accurately regarding their physical appearance became more fashionable, with Alexander the Great's exploits allowing both the spread of Greek culture across several different civilisations and the influence of these civilisations on Greek culture.
Rome had similar ideas to Classical Greece; a lot of their artistic and aesthetic preferences were derived from the Greeks, especially regarding imperial figures. A fairly decent example of this would be Commodus, who was an emperor very concerned with his physical prowess and ability. To cultivate this image of himself as a kind of athlete-cum-warrior, he associated himself with Hercules, the legendary strongman hero of myth. Many statues that he commissioned of himself depict him dressed as Hercules, with Hercules' recognisable club and lion hide. Many contemporaneous accounts do suggest that Commodus was a physically impressive man, but of course there is always the question of exaggeration, self-perception, propaganda etc.
Romans were also, generally speaking, more concerned with individual expression, and there is evidence that Roman portraiture in particular (i.e. painted portraits) as well as sculpture of non imperial figures often aimed for a realistic physical representation of their subject. As Rome began to move away from its reliance on reappropriating Hellenic culture (e.g. the idealised face and form), sculpture as an art form also became more physically accurate, and many Roman busts are quite evidently at least relatively true to life; some take the phrase 'warts and all' literally! To take that example of the red haired man with the big nose, if we were a Roman sculptor, we would probably take great care to depict that patch of hair by his temples that doesn't quite grow the right way, and the moles on his chin. He'd expect nothing less.