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I'm a less clever man who didn't read the "car camping" bit until I was done writing, but I'll let it stand anyway, some of it is applicable.
At 7 months it's too young to carry a pack, so you'll be carrying its food and extra water bottle(s). Even if it's full of energy, you should allow for more rests, a slower pace than usual, or have it used to sitting in your pack when it gets tired. Puppies aren't too good at conserving energy and will blow all their energy from the get-go.
I bring a set of collapsible dog bowls, kibble measured out in portion-sized bags (make the portions 25-50% bigger than usual to make sure the dog eats enough), and perhaps those energy gels for dogs in high activity if the dog doesn't want to eat, like if it's too hot for it. Usual treats.
Offer water at every rest, unless there's water near the trail the dog can drink from as you walk.
Paw balm, wound cleaner, paw socks, knowing how to bandage small cuts, and tick/bug spray, that's all I do for first aid. That, and massaging the muscles at the end of the day. It's not very hard to learn, and you easily mix it into regular cuddling.
My dog is a fucking menace inside the tent, but sometimes it's better to have him in there than outside, getting eaten by bugs. I bring a blanket for him so he doesn't scratch the tent floor or my sleeping bag. If you have his puppy blanket, that's great.
If you don't have one already, consider a dog belt. It's so good to have the dog on a leash while you have your hands free, specially if you're moving in difficult terrain. If you have your dog off leash, it's good to teach it "walk behind" so you can climb down difficult areas and lift it down after.
Extra rope is never a bad idea. You can string it between two trees to make a dog run, or to improvise a new leash if the old one breaks, whether of natural causes or through acts of dog.