How did Voldemort get into Quirrell?

He attached himself after Quirrell failed to steal the Stone from Gringott's. Hagrid had removed it from the vault earlier that day. Voldemort was able to attach himself to Quirrell when he failed to wash his hands after using a public lavatory.

Is Professor Quirrell in Slytherin?

He falsified his school records and false-memory-charmed his professors and fellow students. Quirrinus Quirrell was, indeed, a Ravenclaw, according to canon. However spoilers all, the Quirrell we know from the story is actually a Slytherin. The real Quirrell, who we have never met, was sorted into Ravenclaw.

Why didn't Quirrell reveal himself to Snape?

None of his Death Eaters tried to find him. He presumed that all of them had abandoned him, and had found comfortable lives, and he feared that they would betray him to Dumbledore and the Ministry. Snape is the one that confronts Quirrell, so Voldemort would have seen it as proof that Snape had abandoned him.

How did Snape not know Quirrell was Voldemort?

Originally Answered: Did Snape know that Quirrell was hosting Voldemort? It was never confirmed. He told Bellatrix that he did not know because Voldemort did not reveal himself to him because he didn't trust him, and that he only saw a greedy Quirrel. Dumbledore told Snape to keep an eye on him but that is all we see.

Why did Harry's scar hurt when Snape looked at him?

As he was now a part of Quirrel, his face was outwardly projected onto the back of Quirrel's head. This is why he wore a turban and this is also why Harry's scar hurt him when Quirrel turned around. The fact that Snape was looking at Harry at the same time was purely coincidental.

Why did Snape suspect Quirrell?

In muttering his countercurse at the Quidditch pitch and saving Harry, Snape shows that either he already suspected and was under orders to stop Quirrell OR he was simply doing it because of his obligation to protect Harry from his promises to Dumbledore and because James saved him once, or for Lily's memory.