Well, since this issue's already come up once today, I guess I better elaborate my comment.
As it turns out, why may only be used in a relative clause that modifies the noun reason.
In much the same way, relatives with where must modify a place noun, and relatives with when must modify a time noun. While there are a lot of time and place nouns, there's only one reason noun, and it's reason.
Since the structure is so constrained, and so redundant, something is often deleted,
so that a simple tensed clause following reason implies why.
- He didn't tell me the reason why he wore a polka-dot dress.
- He didn't tell me the reason that he wore a polka-dot dress.
- He didn't tell me the reason he wore a polka-dot dress.
Similarly, leaving out the reason, but leaving in why, produces an embedded question clause
(also known as a free relative clause or a headless relative)
- He didn't tell me why he wore a polka-dot dress.
that also does the same job.
Both versions are correct, and both mean "The dog is not running."
Any difference in connotation is more likely to be expressed via context or tone than the placement of the contraction, in my opinion.
(For what it's worth, the dog's not doesn't technically contain a contraction of is not - it contains a contraction of dog is. But I don't think that's what you're asking.)
Best Answer
No, the order "Why do not...?" is no longer grammatical in contemporary English.