Put the crevasse at a high latitude. High enough that the Sun only lit it at the Zenith, for a few hours at most, assuming a small axial tilt.

And yes, Earth has large quantities of water deep in the crust and even in the core.

Organisms living under the surface might have low oxygen or more likely none at all. This greatly limits the number of species that could live there. Mostly bacteria and small organisms but Im no biologist so I could be wrong.

Also, a desert does not need to be hot, as long as water is scarce. Mars is a good example of this. Its cold and the planet will be at the closest its been for 60 000 years the 27th of July, so Its a good time to look at it up close.