Is Desert Sandstone a chemical sedimentary rock?
No, it's not.

The overwhelming majority of the material in the rock is clastic, not chemical. Chemical sedimentary rocks are rocks where (almost) all of the material was precipitated, such as travertine, evaporites, etc.

Cementation is a chemical process. As a clastic sedimentary rock has to be cemented somehow, otherwise it would be a sediment rather than a rock, all clastic sedimentary rocks have undergone some chemical process. This does not mean that they are chemical though.

As diagenesis progresses and the rock undergoes more chemical changes it may become a diagenetic clastic sedimentary rock of even a metamorphic one, but never a chemical sedimentary rock. This implies that most, if not all, of its initial mass precipitated out of solution.