The PoE system should operate correctly with the described wall socket. It matches the network jack design and supports the intended cabling layout. Using the Dream Machine Special Edition with Camera G4 Pro devices should work as planned. Running Cat6 from the Dream Machine to the wall plate, connecting it to another room, and linking cameras via additional Cat6 cables is feasible. With two wall plates in use, your setup appears viable.
It should function properly as long as your overall distance stays manageable. Typically, aim for 100 meters or less, or 326 feet or less.