Distance - Color RGBLED + Distance sensor HC-SR04 (having strange results)

You can use any pin marked USARTx_TX, so you should be ok. I think DMX uses 12v voltage levels, so you need the level converter anyway (which will be on that shield).

To send DMX, all you should need is:

// if using USART1
Serial1.setup(250000, { tx: DMXpin });
var DMXdata = new Uint8Array(512);
setInterval(function() {
  pinMode(DMXpin, "output");
  digitalPulse(DMXpin, 0, 0.1); // send 'break'
  pinMode(DMXpin, "af_output");
  Serial1.write(DMXdata);
}, 50);

I think you could increase the refresh rate if you needed, and you could make life easier for yourself by not sending all 512 bytes.

I haven't tried that though, so I'm not 100% sure if it'd work.