Personally, I would use the Digital desk and plug it directly into the Mac with Logic just to get a feel for everything. Do some research, learn, read, and experiment with your new equipment.
The Yamaha has a USB out, and on their site it says "The 01V96 has been designed to integrate tightly with leading digital audio workstations to create a complete production and mixing environment. Extensive support is provided for Digidesign's Pro Tools® system as well as Steinberg's Nuendo® DAW - full control of mixing and processing parameters, as well as transport/track-arming control and access to editing functions - directly from the 01V96 control surface. There's also a "General DAW" mode that provides compatibility with other workstations." so it should interface nicely with Logic.
Once you get a feel for everything, start adding in your Rack equipment pice by pice. When you're ready, and have a pretty good idea what ADAT is and does, go ahead and throw the Alesis in the mix and see if you like it. It's better to know about something and not use it as opposed to trying to use something you know nothing about.
I think I would A/B the HD24 and the Yamaha mixer to see if you can hear which has the better A/D conversion and overall sound. The dynamic range specs favor the Yamaha, but not by a lot.
As you do these tests, you will learn to use each piece and get a feel for its strengths-weaknesses in your particular recording situation, and the versatility in your context may determine how to use each.
If I remember correctly, the USB on the 01v96 does not transmit any audio, and is only used for MIDI data and for using the Yamaha StudioManager software.
If you have the ADAT card in the Yamaha desk, then you can take two adat outs from the desk into two adat ins on the HD24 and vice versa. With some setting up in the routing part of the Yamaha desk, you can record on ch 1-16 on the desk into the HD24, then change to layer 17-32 on the desk and playback the HD24 and mix on the desk, or copy everything off the HD24 and drop it into Logic.
If you want to go straight from the desk into Logic, you will need something like a M Audio Profire Lightbridge or Presonus Firestudio Lightpipe