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You should be able to apply it once your dog is dry. I generally recommend applying any topical flea/tick medication at night/bedtime, so it has time to dry and spread over the body by the morning. You are less likely to pet the dog when you are asleep and don't need to worry about it.
I guess it depends on where you are petting him. Not trying to be facetious, but if you spot it on down his back, then you can pet him on his head/neck, stomach, sides etc. Just avoid petting over the wet product.
I wouldn't be too concerned about an accidental touch. Have your kids wash their hands well after petting him. I think it would make the most sense to put the Advocate on your dog after your kids go to bed. It should be dry by morning and then you won't need to worry about it at all. If you plan on bathing him tonight, then I would apply it tomorrow night. Unless you have horrible fleas that you are battling, he should be fine for that 24 hour wait.
I think that should be fine. I don't think you need to wait 24 hours before petting him. Overnight should be plenty of time.