Concerning Quirrell - I think the most logical explanation is that he was teaching at Hogwarts before book 1, but was not teaching Defense Against the Dark Arts. I see nothing in the books to contradict this theory.
Concerning Severus Snape, my theory for a long time (I’ve probably posted this on the boards already, and several people have posted it in this thread also) has been that Snape was in love with Lily Potter. But now I have a different, secondary idea. I’m going to expand on this because I know you all are dying to hear what I have to say.
First of all, there is the theory that Snape has managed to totally fool Dumbledore all these years. Sorry, that theory is rejected out of hand. Snape was once a Death Eater, but no more. (By the way, I think this might explain his desire to teach DADA - he might have thought that either Voldemort’s curse is gone since Voldemort is “dead”, or else that the curse would not apply to him since he has the Dark Mark and is skilled enough to fool even Voldemort into thinking that he is still loyal to the Dark Lord.) To answer the OP, Snape went to Hogwarts on Voldemort’s orders, but stayed voluntarily.
So we have put aside the idea that Snape has fooled Dumbledore. I think that any explanation of Snape’s actions has to start from this point (from Half-Blood Prince, chapter XXV). This is Dumbledore speaking:
Why would Snape be so remorseful?
a) He was a Death Eater but opposed to killing “innocent” people on principle? I don’t think so, a Death Eater should not be surprised by Voldemort committing a murder.
b) He would be sorry that Voldemort wanted to kill James Potter? Unlikely, many people have mentioned how Snape hated Harry’s father. However, and this is something that just occurred to me recently, we must not forget that Harry’s father once saved Snape’s life (as described in Prisoner of Azkaban: Sirius played a “joke” on Snape and told him how to get to the Haunted Shack, and James went after Snape and brought him back before Lupin, in his werewolf form, killed Snape.) And as Dumbledore mentioned to Harry at the end of Prisoner of Azkaban, saving a wizard’s life creates some kind of obligation, in particular Wormtail / Peter Pettigrew now has a debt towards Harry since Harry stopped Lupin and Sirius from killing Mr. Pettigrew. So Snape’s remorse would be related to the debt he owes James Potter.
c) He would be sorry that Voldemort wanted to kill Lily Potter? Yes, if he had been secretly in love with her. This was my original and still favorite theory. In Snape’s Pensieve memory, he insults her and calls her Mudblood. That is somewhat understandable since he was so angry at the time - he probably would be saying things he didn’t mean. Let us not forget that in the memory Lily Potter was defending him. Horace Slughorn tells Harry “no one could have known your mother and not loved her - she was so smart and funny”. Finally, in one of the books (Prisoner of Azkaban again?) Aunt Petunia reveals knowledge of Azkaban being the wizard prison, and when Uncle Vernon is surprised that she knows this, Aunt Petunia says “I heard that awful boy telling Lily about it.” Who was that awful boy? One might assume James Potter, but could the “awful boy” have been Snape, and could Snape and Lily have had a friendship?
My first choice is that Snape loved Lily, second choice is Snape’s obligation to the wizard (James Potter) who saved his life. This explains his change of heart and why he left the Death Eaters.
Now, there is obviously a reason for Dumbledore’s trust that has not been revealed to us yet. (Notice the significant pause in Half-Blood Prince, chapter XXV when Harry Potter asks Dumbledore “why do you trust Snape?”) The answer is (I think I’ve probably also posted this already on the boards): Snape, to redeem himself and assuage his guilt, made an Unbreakable Vow to Dumbledore that he would protect Harry Potter. This is why Snape has always watched over Harry: saving him during the Quidditch games in Philosopher’s Stone when Quirrell/Voldemort was trying to kill Harry, following Harry into the Haunted Shack in Prisoner of Azkaban, searching for Harry in the forbidden forest in Order of the Phoenix when Harry and Hermione escape from Umbridge by putting her in the path of the centaurs, preventing a Death Eater from harming Harry during the final pursuit on the Hogwarts grounds at the end of Half-Blood Prince.
I shall be proved correct on 21 July 2007.