I'll be sleeping in my car and have a camping toilet with pop-up privacy shelter instead of in an RV. Is this allowed here and is this a safe option for a lone female traveller? (Trying to get across the country and avoid planes)
I don't see why not. We always patronize the places we stay and you can always do the same and ask, but I don't think they'd have any problem with it. You can eat at the Visitor Center in the park or right in Holbrook. They also have teepees there and we saw tent campers so they may ask you simply to sty in that area but I would not worry at all about safety.
BTW, don't know how far you're going or if you'll be as far south as Las Cruces but the Scenic Overlook off I-10 is very nice and completely safe for car camping.
I'd call them directly regarding the potty situation, but I don't THINK it would be a problem. As far as safety goes, there weren't many other campers there, but it was really wide open and highly visible, so I think you'd be fine.
I think it would be safe. I don't think a tent or car would be a problem, but during my stay there, I only saw RVs. Give the lady that runs the gift shop a call to check current conditions. She's friendly and helpful. No facilities there other than a picnic table though, so you'll have to be self-sufficient.