An interesting reinterpretation of Wolverine's mutant healing factor occured in Jason Aaron and Ron Garney's story " Get Mystique ", where one of his strategies involved faking his death with a car bomb-- which, thanks to his regenerative abilities, allowed him to make it even more convincing by blowing himself up along with the car. Logan's inner monologue explained that people assume that he doesn't feel pain because he can recover from these massive injuries, whereas in fact the reverse is true-- not only does he feel pain, but the trauma sticks with him psychologically. With everything Logan's endured in his hundred-some years of life, the worst of which being the Weapon X procedure that forcibly bonded metal to his bones and claws ( something which gives him continual heavy metal poisoning, but does not debilitate him thanks to the healing factor ), the pain he feels is as chronic as it is severe, and he has had to accept that nothing will remove the pervasive discomfort.
This is an interesting wrinkle because it makes Wolverine a chronic pain sufferer, and he's far from the type of personality that most in Western culture would associate with chronic pain. At his most simplistic, Wolverine is a superhero who collects every adolescent male power fantasy cliche'. When he's not out chopping up bad guys with his claws, he's smoking cigars, drinking beer, tooling around with motorcycles, and bedding women with his ( literal ) animal magnetism. He endures a lot of suffering from being stabbed/shot/poisoned/blown up/steamrolled/etc ( writers get more and more inventive with what to throw at him for this reason ), but he always heals from it and comes back swinging, a superhuman metaphor for the all-American notion of " playing through the pain ". These are not the acts of a victim bedridden by physical and psychological suffering; Logan can't even be seen as an abuser of pain-relieving drugs, since his healing factor nullifies the effects of all poisons ( including the kind that are socially desirable, like alcohol and nicotine; he must just like the taste of beer, for some reason. ).
If Wolverine is in constant discomfort, it doesn't manifest as a handicap, unless being grouchy and temperamental is a disability. But the hyper-masculine activities that Logan engages in on a regular basis aren't necessarily evidence that he can overcome incomprehensible trauma; it's also been suggested that his adventuring is a compulsion to avoid being alone with his thoughts and feelings. Another great Wolverine story by Aaron started out by spoofing Logan's massive overexposure in Marvel comics by showing how tiring it is for him to fit all his adventures and team-ups with other heroes into a given week, but took a serious tone when Logan finally explained why he puts himself through so much stress-- he doesn't want to be alone with himself and his past. The past Logan refers to is the sins he's commit in his many years prior to being a superhero, but the same logic could be applied to the way he feels about his trauma. It's not something he wants to acknowledge, let alone confront.
Yet many Wolverine stories end up forcing Wolverine acknowledge his inner turmoil, even if he doesn't " overcome " it. The prevalence of Japan in Wolverine's backstory is more than an excuse for Logan to fight ninjas or angst over his dead princess fiance'-- to him, it's a place of peace and contemplation, where he can reach clarity through meditation. Logan doesn't specify a religion, and has even been portrayed as an atheist when it comes to the Judeo-Christian God, but he has a strong ( if also abstract ) sense of spirituality. In his quiet moments, Logan looks to ( admittedly not too specified ) Eastern religion, suspending judgment and finding emotional clarity. Logan can never overcome all of his trauma, but he does find ways to cope with it so he can function as a human being, instead of an injured beast.
Which, ultimately, is why Logan is one of the most heroic figures in the Marvel Universe. That he uses lethal force obscures this for many ( not me, but the whole should heroes kill debate is for another post ), but based on what he's endured, choosing to be a superhero carries much more weight. Logan has lived over a century and endured more pain, suffering, loss, and guilt than any other human or mutant. His healing factor ensures that he cannot age or die, but it also means that suicide is not an option for removing his constant trauma. To cope with his pain, he would have every excuse to be a complete hermit living in the woods, or even a heartless mercenary like his arch-nemesis Sabretooth. But he has chosen to protect innocents and fight evil, even though it means acquiring more pain and suffering.