1) First of all there's no sewage return, I know that for sure. Second, the numbers of "regional resources" are slow to update (maybe once a month?) so snap testing is hard to do. I did turn off my water supply to see what would happen. I am sure Power works in the exact same way as Water as far as the Arcology is concerned, but I did not test Power independently. Also, note it seems the Arcology caps out at 1,000 power and 1,000 water in Regional Resources (meaning it only 'sees' up to that point).
The water or power required to run the Arcology MUST be "available to trade" and not in use. It isn't "sent" (or listed anywhere) to the Arcology but the region has have atleast 120 kgal/hr "available to trade" in order to keep Arcology running.
2) I know in order to get the Arcology "activated" and fully built I had to visit each city in my region, so they obviously have some sort of internal clock whether it's built or not AND whether it's functioning. However, switching between all my cities the "Regional Resources" of the Arcology remained the same.
Basically, while you are playing each city its that city's knowledge of available regional resources when its turned "on" and when its got enough resources to function.
As you can see my water producing city doesn't show any water being sent to the Great Work, but it has more than enough "available for sale".
Hope this cleared some stuff up for you!