Your arguments are based on a logical fallacy: argument from incredulity.
You can't think of a reason that God would heal the dog, so you think God didn't do it or God doesn't exist. But, just because you can't think of a reason, it doesn't follow that there is no reason. You would also have to be omniscient to know that there is no reason, and you're not omniscient.
2 big problems with your argument. I think atheists are starting to realize their arguments are based on incredulity.
I don't see it as an argument from incredulity, Skeptic. It looks more like you've artificially narrowed the options here to "Either the god of the Bible healed the dog, or the dog just healed for no particular reason" -- In other words, a false dichotomy fallacy.
There is a nearly infinite range of possibilities here. Perhaps the dog wasn't as sick as you thought it was. Perhaps the person doing the healing possessed an unknown ability to heal, an ability that could benefit millions if it was properly investigated.
And perhaps it was the work of a god-like being, but not the one you thought it was: What if there exists a benevolent being that helps people without regard to their actual beliefs?
The problem with assuming a god did it, though, is that many of us here did try to cultivate a relationship with the same god that you worship. For us, it didn't work. Our prayers were not answered, and for that reason we are justifiably skeptical about the power of prayer. I, for one, would rather explore the natural world for answers because those kinds of answers can be understood and replicated, and thus benefit more than one dog... Or one person searching for a parking spot... Or anyone who has asked the gods for something they could have achieved under their own power.