What equipment do I need to attach multiple external buttons to a Puck?

So if I were to attach a single button to the Puck, I could attach one wire to D1, and one wire to 3V.

Exactly, yes. Then you do pinMode(D1,"input_pulldown"); which makes Espruino turn on an internal pull-down resistor and you can use. D1.read()/digitalRead(D1) to read or setWatch to get a callback when it changes.

If I wanted to attach a second button, which holes could I use? It appears that there is only one 3V on the Puck.

Easiest might be to just thread two wires through the 3V hole, but you can also use another D... pin instead and then do (for example) D30.write(1) to make that into a 3V output.

Which AWG wire size is most suitable for the small holes on the Puck?

I was using 7/32 AWG wire here (7 strands of 32 AWG), but using that link you found, I think 24 AWG is the equivalent.

As long as it goes though the hole it's not a big deal - if you have any old ethernet/USB/etc cables you could just cut those up and remove the wire out the middle for your testing.

Just a note too - for the external buttons it might be nice to buy ones with holes in the tabs on the end, so you can do a similar thing to attach the wires.