Given that coins were being pressed continously, was it possible to people to forge money?
If I was a wealthy merchant who had picked up a few golden treasures, would it be possible for me to melt them into liquid gold and then dip cheaper metals into it in order to coat them in gold?
Yes it was possible, though it wasn't necessarily easy to fool a merchant. Not only did coins in ancient Greece and Rome have standard sizes and weights, thus inducing the requirement to match the same size/weight ratio of a 'real coin' to trick any merchant worth his salt. Standards shifted over time and between issuing states, of course, but not to the degree that a merchant couldn't keep up with that.
A more common tactic was to /clip/ a coin. Because early coins tended to vary somewhat in practice, and typical measuring tools weren't accurate enough to notice a very small reduction in mass, cheats tended to clip just a little gold or silver of the side of each coin they came across. Most legal systems treated this as the same as counterfeiting (as the effective value of the coins was reduced in the process, and yet they were still used to pay.)
Since in Ancient Greece and Rome minting hadn't yet advanced to the point where producing milled coinage was feasible, they didn't have effective countermeasures to clipping. (Milled coinage provides a lot more control over the end product than hammered coinage, thus allowing for the precisely identical coins of the modern day, with edges that have a pattern that would make any clipping immediately obvious.)
So in short, for the would be cheat in ancient Rome or Greece, clipping would be much easier, a fair bit harder to detect, and overall still give them a good amount of additional precious metal to sell.
hi! there's room for more input here, but meanwhile you may be interested in these previous posts on counterfeiting