There's a thread with a long history answering all these questions, but in short...
It's been debated for a while. And no, Dog did not die in Origins, he just gets knocked out at the end of it, and Wolverine leaves him.
I've always been in support of the Dog is Sabertooth theory, just because it makes their rivalry so much more interesting. And by the end of Origins, Dog basically becomes everything that Sabertooth is today, the only thing we don't see is a healing ability, however, there have been multiple theories on this.
1.) Dog's mutant powers don't kick in until later in life (he was only in his early twenties at the end of origins anyways), and his mutant abilities could come into affect later on.
2.) Dog's mutant abilities haven't fully matured. He's already huge, and he's extremely strong (he is shown lifting Wolverine in the air and throwing him into a barn hard enough to break boards off the wall). And he could possibly already have animalistic senses because he managed to track Wolverine into the Canadian wilderness. His healing ability could just be at a low level, where it can heal injuries fast but still leave scars, not being powerful enough for complete tissue regeneration.
3.) Dog gets his healing powers through Weapon X (though I think the first two options are much more likely)
Also, it's all but outright said that Thomas Logan is Wolverine's dad, so it would make sense that Thomas is the one who carries the mutant gene, which would also indicate that Dog has a strong possibility of being a mutant as well.
