Because the inlet and outlet of 2 are the same size, 2 will always be filled last, if at all. Once 5 is filled, it will spill out the edges of the container rather than back flowing.
If the faucet pressure and flow is so big that it spills out without filling the bucket, the spill could reach 4. Given that the walls of that bucket are extremely cold, the water could freeze over the drain, making that the first bucket to fill up.
Assuming there's a floor it'd be 5 first, then 7, then 4 and 6 basically tie, followed by 2, then a million gallons later 3, and it looks like 1 is last because it's a tiny bit higher
even if there’s no floor, they will all fill eventually. it may cause another noah’s ark situation, but what matters is that all the containers will be filled.
It's gonna spill out and assuming there is a floor and it's not an infinitely bottomless pit, the water will hit the floor, and the water level in the floor will rise. Keep in mind the very first thing I said was "assuming there is a floor"
I think 3 would fill up sooner than waiting for millions of gallons to flood the room, once 2 is full it would over flow the areas to the left and right of it, and assuming 1 is a little higher, 3 would start filling up at that point.