Nice lights. Are they just up there with the adhesive that comes applied? If so, I wonder how well they will stay up there with all of the vibration.
As far as turning on and off the dewalt router, I don’t see a reason why you couldn’t use a relay since the stock power solution is only on/off. If I was going to do that, I would get a relay that is totally overkill and use the small relays you have to trigger the big one. So it would go ramps board –> small DC relay –> big AC relay. Your best bet is to wire it into an outlet and proper enclosure for the outlet. That way you don’t have to cut the chord to the router and could upgrade easily. It is better to go overkill and safe than something that is borderline sketchy, especially since it looks like you are in a school environment.
We do something similar at work when we get bench tools or drill presses. We use foot switches to trigger outlets wired to extension chords. That way the tool only turns on when you turn on the power switch AND have your foot on the switch. It has saved me a few times while drilling dense plastics. If the bit catches and starts flinging the work piece around it is super easy to just step away and let it turn off instead of reaching up and hitting the ESTOP button.