Let's talk tires.

I'm preparing for a trip that's going to involve a decent amount of driving on sandy tracks (Saharan "piste") and beach. I'll be driving a small car with a total weight of maybe 1150 kg fully loaded.

Three questions--

My understanding is that I want less aggressive tread (like picrel) rather than knobby in order to avoid digging in. Correct?

I would like to get something with a heavy duty carcass (like a commercial van tire) for puncture resistance, but I've also heard that such tires are more likely to de-bead when aired down. Is this something that I need to worry about at 1 bar of pressure on the sand? Any other downsides to going with XL or commercial tires?

Narrow vs wide for tires? At a given pressure, they're going to have the same area contact patch, so width just determines shape of the patch. Am I right to think that narrow will be better than wide in order to reduce resistance due to "bulldozing" effect?