We don’t know exactly -what- Reaper’s condition is, but it’s likely to be unpleasa

We don’t know exactly -what- Reaper’s condition is, but it’s likely to be unpleasant at best. It’s heavily implied that his body is in a constant cycle of decay and regeneration - essentially he’s permanently stuck in the process of dying and being revived. And even though he’s able to control the fact that his body dematerialises from time to time, it wouldn’t be much of a stretch to say that he’d be feeling more like a ghost than a man, a shade from beyond the grave. He might very well be in constant pain, although we don’t have any direct clues that point towards it, but even if he isn’t in physical agony, we can’t overlook the damage that everything he’s been through has done to his mental health.

No wonder he’s so pissed off all the time.

If we put aside the idea of Reaper as Darth Edgelord for a moment, there are a lot of clues that point towards how he sees himself. The most obvious one is the way his attire is styled after the classic image of Death as the reaper of souls - even his Mariachi and El Blanco skins follow this theme with their calavera masks. He’s incapable of dying, dissolves into ‘shadow’ at will, and expresses a hunger to kill that goes beyond mere psychopathy. And then there’s his voicelines. All teenage angst and Hot Topic jokes aside, they might well be an indication of how Reaper truly sees himself. “I am Death.” “This is my curse.” I don’t think Reaper considers himself a living thing anymore, whatever the scientifical facts might be.

As for pleasures of the flesh - if he still has any interest in them (and provided he’s still physically capable of experiencing them), it’s likely to be very, very low on the list of his priorities. Reaper is clearly driven by a near-compulsive need for vengeance - for a real or imagined slight, we don’t know. In his own words: “Revenge is the only thing left for me.”
The most positive we get to see him is when he lands a successful kill, or when he’s voted best player of a match, and he comments with a dry “Finally some recognition.”
While I wouldn’t rule out the possibility entirely, I’d say it’s very unlikely that Reaper would engage in gland-to-gland combat, unless it somehow served his quest for vengeance, or if it particularly stroked his ego, seeing as how ‘recognition’ is so important to him. Or if someone managed to bring out whatever is left of Gabriel Reyes in him.

Finally, I’m not even sure that in-universe, Reaper is aware that Jack Morrison lives. Noone seems to have connected him to Soldier: 76 so far. I’d argue that if he found out, Reaper isn’t so much interested in haunting Morrison as putting several bullets into his head - the quicker the better.