It at least removes water from the local water table. You pump water up from the local water table to run your evaporative cooler. The water evaporates. But

It at least removes water from the local water table. 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.