Quest 1: First option; have him actually participate in combat. He doesn't need to be especially powerful, just enough to make us believe him when he says he can handle himself. Or he does that thing followers sometimes do, where they draw their weapon but don't actually fight (tbh I'm still not sure if that's bug or not, but it drives me nuts every time... so, fitting for Sheo).
Second option; he did say he might look at stuff while you were fighting, so... have him do that. Have him completely ignore that combat is happening, and instead maybe he looks around curiously, or wanders off to examine something. It'd be very different from anything we've seen from followers before, and it would help reinforce the idea that he's not even slightly worried about his own safety.
Third option; have him enjoying the show, or looking bored whenever you enter combat. Just, anything other than literally cowering.

Quest 2: There's no point in the quest where you engage in combat while Sheo is actively following you, but he does sprint through a field of bandits that will try to kill you if they see you. Presumably, they wouldn't just ignore the old man in expensive-looking clothes running by. Just put one or two dead bandits at his feet when you catch up to him at the house.

Quest 3: He's only just starting to feel the effects of the poison, and he's furious, so his animations should reflect that. He might be unarmed, but apparently the "thrill of the chase" can temporarily clear up the effects in the next quest, so I think rage and adrenaline can absolutely sustain him through the end of this one. He should look ready to start swinging if he had to. Lucky for him he can't be targeted by NPCs.

Quest 4: He's feeling bad enough now that he needs to sit down after exerting himself, which makes it weird when his posture doesn't always reflect that ("thrill of the chase," indeed). It's also a bit odd that he doesn't actually follow you at any point during this quest, especially since he does so during the next two, when he's supposed to be feeling even worse. Maybe instead of just appearing from nowhere when entering Othuk's hideout, he should tag along. At this point, it's finally started to sink in that his life is in very real danger, so if he's going to start cowering when you enter combat, this would be the time.

Quest 5: He's weak enough now that with his staff depleted, he apparently doesn't have the strength to carry it while you're running through Enduum. At least, that's the justification I came up with. When he handed the staff back to me before leaving for Enduum, it was immediately, immersion-breakingly obvious that it's because I'm the player and I need something to do. The thought that maybe he's too weak to carry it around came around later, after thinking about it. I'm not even sure if that was the intended reading or not. I think at this point, he should switch from the regular walk/run animations to ones used by sick or wounded NPCs.

Quest 6: Mostly the same comments/complaints as the previous quest. He's supposed to be very weak, and the only reason he hasn't collapsed yet is because of the staff, and the scraps of his power you've been collecting along the way.