Coinage production methods varied over time and from place to place. Like many ancient crafts, the art of coin-making would have been kept as secretive as possible; doubly so for coin-making, given the potential for abuse by counterfeiters if the secrets became widely known. So the wheel was no doubt re-invented countless times, and different ways of obtaining the same result were employed, as lost secrets had to be re-discovered.
Casting blanks was clearly one way of making them which certain regions and times employed. Many Ptolemaic and Seleucid bronzes, for example, have a trapezoidal cross-section - clearly indicating from their shape that the blanks used to make them had been cast in bowl-shaped moulds.
Some large bronzes have clearly had their rims and faces turned on a lathe prior to striking, as they still preserve some of the lathe-marks. Lathing the rims made the resultant coins rounder and lathing the faces allowed for a clearer impression to be struck from the dies.
Drop-casting seems to have been commonly employed, especially for silver and gold coins, where a measured amount of molten precious metal was tipped into cold oil or water, freezing it into a more-or-less spherical shape which would then be squeezed flat by the dies.
I recall an episode of
Time Team where they did some experimental archaeology in trying to make ancient-style coin blanks and coins by various methods. Chiselling blanks off a bar of metal, cob-style, simply didn't work well for bronzes - the cut was hard to make cleanly and the resultant effort to make the coin even approximately round was far too time-wasting to be viable for the mass-production of coins. They also found that heating the bronze blanks prior to striking was pretty much essential for bringing up the high relief found on ancient coins.
As for the dies, very few of them have survived - again, as state secrets with a high potential for abuse they would have been jealously guarded and deliberately destroyed once they got too worn, rather than simply being thrown into a rubbish tip where counterfeiters (or later archaeologists) might find them. But yes, all ancient coins were made using hammmer-and-anvil dies. Hand-held, for the most part, although some Roman dies were apparently employed in a hinged box-like apparatus, in an effort to keep obverses and reverses aligned consistently - a concept lost during the Dark Ages and not re-invented until modern times with the concept of machine-struck coinage.
Because of the lack of surviving records and artefacts, there is still much about ancient coin production that we simply do not know. For example: many ancient bronzes, from places separated in both time and distance, bear curious dents in their centres which were placed on the coin blank before it was struck; the debate, about what these
centration dimples actually were caused by, still rages.
Don't say "infinitely" when you mean "very"; otherwise, you'll have no word left when you want to talk about something really infinite. - C. S. Lewis