But from the perspective of everyone else, Lily and James are dead because they were betrayed by their secret keeper, Sirius was the secret keeper ergo they are dead because of him and he must be the spy. This is compounded by his "murder" of Peter and the 12 muggles and him standing over Peter's remains laughing his head off.
There's not much anyone would question.
Hagrid even comes up with an explanation for what Sirius is like after James dies - he says Sirius was all white and shaking and he comforted him, thinking he was upset about James when really he was grieving his master's downfall. Sirius asks Hagrid to give him Harry, and Hagrid supposes that if he had Sirius would have thrown Harry off the motorbike at his own convenience, but that when Hagrid says "no" Sirius hands over his motorbike as it will be too easy for the authorities to trace and he has to make a run for it.
Everyone thinks it was Peter who tracked Sirius down, not the other way around - so Sirius was on the run, Peter found him, Sirius killed him (and 12 other people and then stood there laughing).
From the information everyone has, everything makes sense or can easily be explained away. As the truth is that one of the dead heroes was actually the spy, no one is going to guess that. It would be a dishonour to Peter's memory to even think it. So even someone who wanted Sirius to be innocent couldn't possibly come up with another version of events that made sense.
With the information he has, Remus can either stand by Sirius or stand by the memory of James, Lily and Peter. But he can't do both.
As for Lupin not giving up the entrance to the Honeyduke's passageway - he is presumably going with Ron's assessment of the situation: that Sirius cannot break into Honeydukes with all the dementors floating around Hogsmeade therefore the passage is useless to him. Therefore no one needs to know about it - even if it's still there (after all, the fourth floor passageway behind the mirror has caved in since the marauders' time). There are actually 7 passageways out of Hogwarts, and Filch knows about four of them, maybe Lupin is choosing to believe that in the past twenty years some of the other passages have been found - or perhaps he never knew Filch didn't know about it. Lupin is a character who runs away from his problems and buries his head in the sand, it is a core part of him, he will be coming up with all sorts of reasons why he doesn't have to pass this information on.
The truth is, he could tell Dumbledore about the passageway and even hand over the map without revealing to him the truth about Sirius being an animagi and without having to admit to breaking any rules (it's not against the rules to know where the passages are, or to draw a map of the school). Cowardice explains why he doesn't tell Dumbledore Sirius is an animagus - he doesn't want to lose Dumbledore's trust - but it doesn't explain the other secrets he keeps... But it's love. He still loves Sirius even though he hates him, and though his head might accept that Sirius belongs in Azkaban his heart will not allow him to be the one who passes on the information that results in his best friend being thrown back to the dementors. So he keeps quiet and desperately trusts to Dumbledore and Hogwarts to keep Harry safe, and prays someone else will capture Sirius without him having to get involved.