Well, if you think of this scientifically rather than science fictiony then we already have the answers.
Dogs have evolved on their own for millions of years, with man purposely interfering in the process for only a few thousand. If Great Danes, or GSDs, or even the wily Poodle could exist on their own without benefit of man, then they already would have.
The influence that man has had is not to make dogs more successful with out us, but to make them more dependent on us.
The most compelling arguments I have seen that explain the rapid evolution o modern dogs is that it is not evolution at all, but rather devolution. Dogs are dumb wolves. Certain dog breeds were developed to emphasize the tracking instincts, certain the chase, other the kill; with all other existing instincts de-emphasized.
Now, if you allow for us leaving huge heaps of trash and empty cities rather than a pristine earth, it is possible some breeds (my guess small sneaky ones) might survive until the trash ran out. Otherwise, the world will be left to the cats, the wolves, and the pigs, who do just fine without us.
George