Harry has ALWAYS believed the worst of Snape.  Harry now respects the man, cares about him and trusts him as an ally against Voldemort.  But his underlying assumptions about Snape's character haven't really changed and he has no problem believing Snape capable of killing in cold blood, especially after what Voldemort put him through.  

Also, Harry is out of his depth because he hasn't had to deal with this sort muddied issue before.  His challenges have all been blackand white: stop the evil guys from doing something despicable and/or save his own life.  The closest he's ever come to dealing with shades of gray was in Snape's Worst Memory when he had to come to grips with the fact that James and Sirius weren't always perfect people.  And the kid was traumatized by it.  That however was an incident from the past before he was born and since Sirius died soon after, it isn't something Harry has to deal with in his life.  Snape is. 

Snape told Harry that there is a difference between being of age and being an adult.  That's true and it has little to do with bravery.  It's about understanding others and oneself.  Though Harry is mature beyond his years in some ways, he is still very much a child in this department.  Snape presents Harry with an invaluable opportunity to test his own presumptions and prejudices and in doing so, to grow up.