Is it actually possible to make carbon-copies of individuals this way?
It is almost certainly possible to clone humans. What is far from clear is whether you'd be able to make "carbon copies", because a babies come with fairly minimal mental facilities pre-installed (notably the ability to learn, especially language) and everything else gets picked up as they grow. You cannot practically replicate the childhood, adolescence and early adulthood of any particular human genius, so what you'll mostly have is an expensively produced relative of some famous figure with a frankly terrifying amount of expectation laid upon their shoulders which seems mostly like a recipe for traumatizing children (and don't get me wrong, if this is your intent then a setting that has a production line for depressed and rejected clones forever living in the shadow of their forebears and possibly billed for their failure to deliver sound like an excellent scifi dystopia).
People who have accomplished amazing achievements in intellect (Nobel Prize winners), physicality (Olympic Gold Medalists), beauty (decorated beauty queens or supermodels), or other major cultural achievements (famous humanitarians)
Are you aware of the notion of Nobel disease? Or perhaps of the long, long list of female scientists whose work was stolen by their male colleagues? Are you certain you're not selecting for people who got lucky, and who were supported because of the position they held in their society at the time thanks to their gender or skin color or who they were related to? Humans famously "stand on the shoulders of giants", and even the circumstances that brought about success in the original might not rise again, even if you could perfectly replicate the person who achieved them in the first place.
Beauty and athletic prowess at least seem somewhat more likely to be replicable given a suitable set of donor gametes and a healthy upbringing, but other than the fame factor I can't help feeling you'd get a better return on investment by ensuring that more children got a healthy and supportive upbringing so that their natural aptitudes were able to flourish, but I guess no-one wants to read about boring scifi utopias.
Recently, I read an article about stem cells being transformed into sex cells (both sperm and egg). It made me realize that it could be possible for instance to take a sperm donation from a man, transform his stem cell into an egg cell, implant it inside a woman, and outcomes a baby clone of the man.
Whilst this isn't immediately obvious, autogamy does not imply that offspring which only have a single parent are in fact clones of that parent. Even without going in to the complexity of epigenetics, it is possible for genes to be heterozygous, meaning that the parent can have two different variations of a gene in their DNA, but a particular gamete they produce can only have one of those variations. That means that a random pairing of sperm and egg from a single parent can be homozygous for some of more of the parent's genes. Genetic dominance is complex, but you can end up with a bunch of non-clone children who share a single genetic parent and yet who nonetheless might end up with things like different hair or eye colors, some having a genetic disease but not others, and so on.
For complex traits like "intelligence" it seems like an absolute nightmare to guarantee that any one child made this way would actually reflect their genetic parent's actual genetic capabilities, even before having to deal with the even more complex issue of environment and upbringing. Again though, a fictional (at least for now) setting where someone can be an unambiguously inferior recombinant child of a "great" person in a way that might only become obvious later in their life does seem like an excellent way to introduce some instant drama and conflict.
If you want to make clones (and be aware, these can't be perfect clones without magic!) you'll need to have the DNA of a regular cell from your donors... sex cells alone will not be sufficient.