You pump water up from the local water table to run your evaporative cooler. The water evaporates. But the air was at 10-20% humidity. The water from the evaporative cooler will raise the humidity, a little bit, but not enough to make it rain. It may make it more likely to rain somewhere downwind a few miles, or a few hundred, but not here.

For your second dumb question: At least some of the Great Lakes have at times had issues with water level. (They want enough to allow ships to pass between the lakes.) The upside is, there the humidity is high enough that you're more likely to get the water back in the form of increased precipitation.

Sometimes it's true, but it takes 10,000 years so if you mean "recharges while our current civilization exists" then it's effectively false. This is the case for the Ogallala aquifer which supplies 27% of the irrigated land in the US.

https://en.m.wikipedia.org/wiki/Ogallala_Aquifer

Sometimes it's just completely false because the groundwater exists as a sealed bubble of water put there during the last ice age. That glass of water you just drank? Congratulations! It's 25,000 years old, and we know this through isotopic analysis. Your drinking water was mined and once it's gone it's gone forever.

https://en.wikipedia.org/wiki/Fossil_water

Sometimes recharge can happen but only via snowmelt at high altitudes (and of course Tucson doesn't get much snow).

Sometimes it's completely true. The Edwards aquifer in Texas gets recharged every time it rains.

https://en.m.wikipedia.org/wiki/Edwards_Aquifer

So it's complicated. Generally speaking, deeper groundwater sources take longer to recharge if they recharge at all. I don't know the particular groundwater situation around Tucson.

> why are we building projects that need tons of power and water in the Sonoran desert instead of next to the Great Lakes

Evaporative cooling is effective and cheap where humidity is low. It doesn't work well where humidity is high.

Very little of the water used to grow the product actually ends up shipped. For example with alfalfa (a major Arizona crop) only 0.005% of the water used to grow it end up in the alfalfa. The rest of it first ends up in the ground near the plants or in the atmosphere near the plants.

This is also the case for most high water use industries. For example making integrated circuits uses a large amount of water but very little if any actually ends up in the chips themselves.