Well, getting stabbed in the neck isn't fatal, getting hit in the carotid artery is fatal, getting stabbed through the spine in the back of you're neck is fatal, but you can be stabbed in the neck and survive.
That said you can also take an axe to you're head and survive, it's unlikely but given the proper medical treatment you can survive.
Also to people who are upset over the immortal main characters cliche, you can actually survive being shot in the head with a bullet. It really depends on where it hits.
Though, yes I'd say most shots are fatal and I will have to admit the shot that Indian guy took should've gone through both his hemispheres, so he's definitely supposed to be dead or at least brain dead.
My own thoughts about this is that I never took this manga as an assassination manga, if you want an assassination manga I'd suggest Golgo 13. I look at this manga as the relationship between a quirky couple who's quirks are that one's an assassin and that the other's a bounty hunter.
I've always wanted to see a manga with that kind of relationship dynamic, a sort of Mr. and Mrs. Smith manga, so I'm pretty OK with the main character's being immortal.