Ok, I was about to throw in the towel but went back and reread your last 2 posts....I read the first of the two wrong. So let me regurgitate the info and see if I have this right. You have a 24 volt transformer powered of course by 110. Pictures 2 and 3 are of the transformer. Picture 1 confuses me still. Does it shows 2 hots and 2 grounds, or one hot, one neutral, and 2 grounds? The wire colors are not normal for 110 volt. Is this location a switch, the transformer location, or the old DC location, and is it 110 or 220? Picture 4 is obviously a switch, but it doesn't have enough wires for a 3 way switch, at least not a normal 110 volt 3 way light switch. And the wiring is not typical low voltage wiring, it looks like 14 guage that would be used on 110 volt.
Now, with that said. If the current relay triggers 110 voltage that powered the DC, then the easiest thing to do is run your 220 volt wire to a location close to where the old DC plugged in. Take that 110 volt line and use it to trigger the new relay (trigger terminals are generally on the sides of the relay with the power wires being switched running through the middle-follow your relay wiring instructions for this) Run your 220 into the relay input, attach the 220 wires from the output to the motor. Ground runs through without being switched by the relay. This way all switches work as they have been working.
Next question: How did the original DC hook up to power? Was it hardwired or did it plug into a wall outlet that was controlled by the relay and switches? If the former, you will have to cut the wire from the 24/110 relay and wire the wireless remote in there. If the latter, get an electrical box for the new 220 relay (that may come with the cyclone now?) wire your trigger wire where it comes out of the box and can plug into the outlet the original DC was plugged into. This way, you can unplug the trigger wiring, plug it into the wireless remote box, and plug the wireless remote box into the receptacle.It will allow your current low voltage switches to become the master power on/off, while your wireless remote acts as the cyclone on/off. If a low power switch is not on, the wireless remote will not operate the cyclone. This would keep the remote from picking up a weird signal and turn the cyclone on when you are away. I wired my system this way only I have only one light switch that is 110 that triggers a 110/220 relay to run the cyclone. I have never purchased the wireless remote as this setup works just fine for me. But if I decide to, it will just be a mount, unplug trigger wire and plug into the wireless remote box, and then plug the wireless remote box back into the outlet.
I hope this is clear. My suggestion is to get an electrician to check through everything. I am not an electrician, so I am only giving you information on what I think is right. Best to use a professional to make sure it IS right. If you decide to do this on your own, at least talk to an electrician about what you are doing. Maybe you have a friend or acquaintance that is an electrician. Run this past them. If you are inside city limits, call your local building permit office. Chances are you will need a permit. Also talk to your Insurance Agent. It is possible that if you, or an electrician, do any wiring without a permit and it burns down the house, you won't be covered on your insurance. Don't make the mistake and find out after the fact.
Good luck and let us know how it all turns out! Jim.