Okay I live in Phoenix too, as do MANY people and it is just fine for the dog to go outside to go to the bathroom. Would I recommend leaving your Boxer out all day? Absolutely not, but 3-5 times a day to go to the bathroom and spend 15 minutes outside is just fine. I do not have a dog door either, I let my dogs out every 3-4 hours (sometimes less or more depending on the day, what I'm doing, if I'm home, etc) and they go to the bathroom and come back in.
I'd really encourage you to throw down some grass seed in your yard water it a couple of times a day and you'll have a lawn in about 2 weeks. This is what we did, we have a 1,500 sq ft yard and it cost us about $40 to put grass in out backyard this way. We put down winter grass in the winter and summer grass in the summer. Once you plant the summer grass once you never have to do it again, winter grass you have to do yearly if you want it to grow in the winter (or you don't have to the summer grass will just look dead (dormant) no big deal). I've lived in Phoenix almost 5 years now and have had 3 dogs total since living here (all boxers) and it's just fine for them to go outside to go to the bathroom.
I'd ditch the "litterbox" and teach your dog to go outside like it's natural for dogs to do.
I'll also add, we had only dirt for about the first 6 months of living here and we also have hardwood floors and carpet mixed, so I'm not sure what the point is about you having dirtin your yard and hard wood floors? You clean the floors when they get dirty and wipe the dog's feet off if it gets muddy in the yard (although if you throw some seed down problem solved). I used to just take a dust mop and spend 5 minutes a day going around and that got the dust from the dirt up off the floors easy enough.