Should I make payments for a property purchase directly to a developer or, preferably, to my agent?

Yes, you can transfer payments directly to your developer. Most of the real-estate developers are genuine and reliable. However, there are still quite a few that are not very efficient- and reluctant parties to deal with. If you want to ensure smooth proceedings you should at least engage a professional real-estate agent that is familiar with the market and able to assist you in case of any problems. Checking the track-record of both the developer as well as the agent that you are dealing with is advised.