Depends on where you are going, some places just suck and NM has plenty of them, I went to Colorado last week and a few places I didn't set up the tipi just because the only nice and flat places to pitch was powder dirt, I hate it when every move you make in the tipi creates 5 minutes of inhaling dust.
I generally don't use a tent when I'm camping in this state. I have a cot and a tarp if it looks like rain. Sleeping out is one of the nicest parts about being in NM, from the eastern plains to the high Sangre de Cristos. There have been some biting insects this summer - mosquitoes and noseeums (Family Ceratopogonidae if you want to get technical) around water sources. On the other hand, I was camping up around Red River in early June and never experienced a bite.
The dust would definitely be an issue in a floorless shelter. pack your tyvek or whatever you use for a floor.
Driving to Albuquerque to pick up son and then north to fly fish and camp between sante fe and Taos. Just playng trip by ear. Hopefully most of July fourth crowd will be heading home by the sixth. Really hate spending 350 for trout guide but probably will. Usually well worth the money for new places.