I'm just delurking here because evolutionary biology is my field and there are a couple of misconceptions that are bugging me a bit.
Basically, the definition of a species has to do with barriers to gene flow. Postzygotic isolation--that is, never being able to produce offspring together--is the gold standard, yeah, but there are also good biological species that can theoretically create offspring in really unusual conditions--like captivity!--but never or almost never do in the wild. Your king snake idea is an awesome example of this! No genes are being passed back and forth in the wild because king snakes generally eat corn snakes instead of mating with them, so you have reproductive isolation as a consequence of behavioral traits. So they're still good species even though you can, with careful finagling, get a fertile F1 hybrid, because in the wild they never reproduce together and gene flow between corn and king snakes is nonexistent.
Speciation is also a process, not an all/nothing boundary--there are honestly almost none of those in biology. You start with two populations of the same species, with no barriers to gene flow, and you end with two species where no genes can flow back and forth, but there are some intermediate steps along the way. Behavioral isolation like the king snake example and other forms of isolation that happen before mating can take place usually happen first, which is why we have so many examples of good species that can produce occasional hybrids under unusual conditions, like captivity.
Dogs and wolves, if allowed to live in proximity to each other, will still freely interbreed without human intervention. Therefore they are not good species. (For example, feral dogs and wolves interbreed very commonly in Italian garbage dumps, where both are known to live without much in the way of human intervention.) This is not true of, say, dogs and coyotes--yes, rare hybrids exist, but there are enough natural behavioral barriers that they stay very rare even though coyotes and dogs often do live in close proximity to each other.
With respect to animals in different genuses being able to produce offspring together, well, while species are a natural unit of biodiversity (sort of the way that days, months and years are a natural unit of time), taxonomic classifications above that like genuses, families, and orders actually aren't (sort of like weeks). We sometimes use them as a shorthand to help us conceptualize the evolutionary tree of the species we're talking about, but they're largely a holdover from an older, Linnean school of classifying life. Most biologists today think primarily in terms of species.
Does that help to clarify anything?