Connecting HBP with what JKR said about the third movie

[Jo]

No: HPFGUIDX 133355

JKR told us that when she saw the Azkaban movie, there were scenes 
put in that would appear to have been place there on purpose, because 
they fit so well with the story she is unfolding.  I can't help but 
think of the scene in the movie where Lupin is talking to Harry about 
his parents.  He tells Harry that his mother could "see the good in 
others even when they couldn't see it in themselves."  Slughorn also 
tells us over and over what a good Potions student Lily was. There is 
not doubt that "Prince Severus" was also an excellent Potions student 
(judging from Harry's luck with his book and Snape's teaching 
field.)  

I was just wondering - 

Did Gryffindor and Slytherin have Potions together during the 
Marauders' era?  

Could Lupin include Snape as one of those in which Lily saw 
the "good?"  

Is it possible that Snape hates Harry so much because he is James's 
son instead of his own? 

Did Voldemort promise Snape that he would kill James and Harry but 
spare Lily? 

Does Snape see Harry as responsible for him having to kill Dumbledore?

Just wondering ---

Jo