It shouldn't be losing power.. can you try a different Ethernet cable? could be shorting. or even a different port on the switch? when it's up, can you run 'show inline power' to see the power draw and PoE class assigned to the AP-port? What version of firmware are you running on the ICX? i think the current recommendation is 8.0.70c which might help. Can you try some of these and let us know how it goes? Since they are both going offline (do i have that right?) its not likely a faulty AP.
Any of the ports can provide PoE.. i just suggested using different ones in case there was something wrong with the physical port. I've seen environments that were not dry cause some corrosion, or there could be physical damage... just trying to rule it out. And yes, it CAN provide DHCP, but you need to actually configure and enable it so it is most certainly not something on by default. Rogue would be another WLAN that ties back to the same network.. so makes sense but not a problem. i have several rogues in my lab... 🙂
hmm... the "media closet" is in the laundry room. I wonder if that could be the case. Not sure if other ports are powered out of the box though, and still trying to learn how to piece all this together to re-initialize everything