Does this evaporated water stay in the loop and condense out later? Is an extra liter on the front end needed in the line until the first liter falls out of the back end?

What does that mean for environmental impact? Hydroelectric power uses water in the sense that the "used" water is at a lower altitude. What happens once that is done, is what matters.

Depending on what global average means, it seems like that's quite a lot of cycling of evaporation unless they are releasing steam at 800C

Looking up overall water usage, the US uses 27.4Billion gallons a day residential and 18.2 Billion gallons industrial. It surprised me that industrial was lower, but I guess the US manufactures less these days.

If the 1kwh per litre were accurate then judging by this calc https://www.google.com/search?q=(27.4billion+gallons+*365)*+... 37 857 903.3 terawatt hours

0.1% of The residential water use of the US would be enough to cool the entire Electicity output of the world (about 30,000TWh)

(of course, with these things it's easy to slip an order of magnitude(or several) so best check my numbers)

It means different things in different places, depending on where the water was sourced and the counterfactual of what would otherwise have happened to it. For example if you source surface water that evaporates, and the local rainshed feeds that watershed, then it's virtually closed-loop. If you're pumping fossil water out of an aquifer, and it rains over the ocean, then it's the opposite.

I suppose in some desert-ish climate the extra moisture given to the air doesn't rain down nearby but moves with the wind somewhere far away where it rains. So there might be an effect of using locally precious water even if it's a closed loop on a continent level.