There will always be tensions, in my opinion. Whilst Slytherin gets it for being "evil", Hufflepuff gets it for being "duffers", Ravenclaw gets it for being "swotty" and Gryffindor is seen as "dorky". There's a way to put a negative spin on anything. As much as houses give a sense of community and unity, it also divides as there's the element of competition.

I assumed he meant that some Slytherins fought for Hogwarts. I don't see how he'd have known about Snape - no one did, for their own safety.

I think the point was there will always be prejudices, but what matters is how much precedent is given to them. Hence why Harry tried to impress that it didn't matter where he was put.