No. MyDMX still does not support any sort of export, nor will it in the future. Plus, Logic is not a DMX application in and of itself.
You can trigger scenes in MyDMX using MIDI note on commands, which can be triggered via Logic. Also, thanks to the implentation of CC data, MyDMX can also be directly controlled through MIDI. However, it needs to be said(since nobody will search here for it) that you SHOULD NOT under any circumstances run MyDMX and expect to control it from the same computer with a MIDI-capable program AT THE SAME TIME. Doing so can lead to instability problems as well as a straight up crash if not done properly. Both are things you don't want to happen in a live event.
MyDMX does not install itself as a MIDI device or virtual instrument, so with the lack of a MIDI bridge functionality or virtual instrument functionality, My DMX requires that it get its control signal via a MIDI IN port AND must run as the foreground application in order to receive MIDI commands.
You will note that MyDMX is also scene based, and based on what I saw in that video, you're mainly triggering scenes, which is fine, can't say anything bad about that.
If you want to see things more or less in real time, MyDMX has a 3D Visualizer that can pretty much show you what's going on, within reasonable limitations. I rely on this heavily for lighting design projects myself. While not 100% accurate, it does let me know that I am within reasonable tolerances and I can fine tune the rest on site.
Personally speaking, with the way MyDMX is now, if you insist on using it in a live situation with MIDI sequencing and/or pre-recording tracks, you want and NEED to use 2 computers and 2 MIDI interfaces, 1 MIDI interface per computer. You MUST use MIDI as your transport protocol.