Historically large data centers used evaporative cooling. So the water disappeared in sense that it's no longer available in a useful form in it's original locations. That's changing with growing use of closed-loop systems.

A lot of people think it does, surprisingly. People memed too hard about "querying ChatGPT drains lakes" and anyone who contested it was deemed a booster, so it stuck.

no it doesn't but it doesn't go back to where it was taken from either. So it does affect local aquifers.