I don't know who my character is. He put his life on the line for the empire and yet he killed the emperor without regret.
He might've liked the empire as an organization, but just not the leader of it. The emperor was weak anyway, and completely unable to protect his territory against the Aldmeri dominion. So you're in fact very passionate about your loyalty, but only to whoever you really pledge it to (you didn't do it to the emperor in your mind, only to the empire). That oath you swore? F.u.c.k. that, you don't believe in oaths/promises/vows/... anyway!
He became Archmage without casting a single spell.
You're Dragonborn, revered by all Nords. The College is suffering from scorn amongst the general Skyrim populace (especially Nords). What better way for the school to boost its popularity by making the Dragonborn the archmage? About the Psijics, just imagine they probably know something about your future or whatever which drives them to supporting you in your archmage-ness. Oh yeah, and you ... you had to infiltrate the College to get into Saarthal, and afterwards felt obliged on keeping up appearances at the place by not stopping to take on jobs. Those mages could come in handy once again after all.
He lead the companions and is listener for the brotherhood.
You simply have a dark side, which you keep hidden from all other factions. The Harbinger isn't even a real leader to the Companions, only someone with a bit more influence. You
are a warrior, so I don't see any trouble with being a member of the Companions.
He pledges his soul to Nocturnal when his soul belongs to many others already.
I guess your character would love to see the infighting between the Daedra and Aedra after he's died. As do most of our characters anyway.
He is the Dovhkiin and he killed Paarthunaax and betrayed the Greybeards because he was told to. Morals does not come to mind when he is asked to do something, he does things without hesitation simply because he was told to do so. He served all Daedric gods without hesitation.
Well, you're saying it yourself: your character is simply someone who always feels obliged to do everything what people tell him to do, without thinking (some sort of mental disorder). Like an addiction; he feels very restless if he doesn't do that. He's in fact a perfect soldier. Also explains your passionate loyalty to those you deem worthy (or at least worthy enough at that time), why you kept on taking jobs from the College despite not being a mage, and so on. You're not an extremely altruistic character, no, you simply have OCD with following orders and can't decline requests.
Who is my character? Can I explain my actions to properly roleplay with him? Is it even possible to roleplay now with the things I have done? After 100 hours of gameplay with him I feel nothing he did ever mattered.
Well, he certainly isn't easy to roleplay, but everything is possible, as long as you twist it enough. And he'll probably become pretty boring. You could always try to invent some background information about him (how was his youth, what happened which drove him to Skyrim, ...), and generate some character development. Maybe his condition will get worse, and he'll hallucinate and follow orders from imaginary npcs, or the like ...