Initially yes, a tiny bit, but long term no: it will go to a new equilibrium, where over time just as much leaves as enters. Except thanks to the extra water that is being retained locally, the local environment will be greener + more temperate than without the panels. And that new green lung is likely to have beneficial effects further away as well, similar to forests.

I mean that is quite a good question. My argumentation would be, that by melting the ice caps the amount of liquid water on earth generally increases, so keeping some of it in the desert does not seem like a bad idea.

But i am in no way qualified for a climate simulation. If anyone knows more, ping me!