Leave no trace guidelines in mind, I would say no, the dead wood plays a role in soil nutrition and wildlife should not be removed. Besides, It's a desert!

I would assume it is not. More importantly, this area has little or no usable wood suitable for a fire as far as I could see.

But why would you? This area is heavily impacted and just because you can does not make us good stewards of this beautiful area. And yes I am a second amendment conservative.

It would be nice to leave the fragile desert environment in its natural state and bring in your own firewood. Quartzsite sells boxes of really nice stuff for not a lot of money. You can find it a the corner Kuehn

According to BLM regulations it’s ok

Visitors to the forest can harvest special forest products in reasonable amounts for personal use without a permit, such as collecting firewood for use at a campsite Harvesting more than small amounts requires a permit, and in some cases may require a contract. Some desert areas contain salvage-restricted species, such as barrel cactus, Joshua trees and saguaro. These species cannot be taken off public land without a permit.