- Was the playdate synth based around an existing synth?
The synth code is my own creation, a fairly standard softsynth implementation with generators, modulators, filters, etc.
- How does the playdate play audio? Is there a special chip involved?
It's all generated in code, there's no synthesizer or DSP chip involved on the device.
- By any chance, is there a soundfont file around?
We don't have soundfont support yet, but I'd love to add that one day. A while back Takuya ported the TinySoundFont library to Playdate but reported that it didn't perform well on the hardware. One of these days I'd like to take another look at that and see if I can make it work better.
- Is there a way to emulate playdate audio in a DAW?
I'm not quite sure what you mean by that. 
If you're looking for a way to use the sound engine but avoid using code I've seen some tools like Simple tool for checking synth sounds on device that let you explore different parameters, but eventually you'll have to set everything up in code. After that you can trigger notes in code, possibly in reaction to buttons or events in a game, or you can play a .mid file as a sequence.
I know a lot of the API expects that you already know what all of the term mean. If I ever have the time I'd love to have the time to write a full tutorial on how the synth engine works; until then, there's sample code in the Examples folder that might be useful.