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City of Phoenix has mommy & me type classes. There's also My Gym, Gymboree, mall play areas, splash pads, Imagination Avenue, Makutu's Island, Arizona Science center, Jambo Park, The Play Fort, Play Factory, the children's area/story time at the library. Not sure about what parks are lit at night - you may want to check out City of Phoenix's parks site and/or call for that info.
Been there, done that a little one during the summer & I learned to be a little more flexible. You've got a lot of "no's" on your list, which is a disadvantage for you. Germs are everywhere, and he will need to learn how to adapt to being in the car.
In any event, I hope I helped. I know it sucks, but you'll figure it out if you open up your options.