But stop to ponder upon this: what if things had gone differently?
Well, the first thing that came to mind when I read the question was that we would have a lot more insight into Malfoy, Crabbe, and Goyle! I always felt like viewing the story entirely from Harry's perspective was a bit of a disservice to those characters.

What if Harry had accepted Draco's hand of friendship aboard the Hogwarts Express, or shortly after arriving at Hogwarts?
This would also mean that we would have less of an insight into Ronald and Hermione. However, it could also be that Harry just has a much larger friend group as a result.

How might that single decision have changed the subsequent story? For instance, would Harry have still ended up in Gryffindor, or could his relationship with Draco have influenced his choice between Gryffindor and Slytherin Houses?
His relationship with Draco would have influenced his choice. Remember, it was Ronald in his ear that convinced him to choose Gryffindor. Ronald told him "there wasn't a single witch or wizard who went bad that was not in Slytherin," with Ronald not by his side to say that to him, his perception of Slytherin is not tainted immediately before he puts the Sorting Hat on.

Could Harry's friendship have changed Draco for the better, or would Draco have pulled Harry toward different values?
As I said, this depends on what new insight we get into Draco. Draco only treats Harry the way he does in canon because of Harry's rejection of him. Take away the rejection and, from Harry's perspective, Draco is immediately better.

If Harry had been Sorted into Gryffindor regardless, would his and Draco’s friendship bridged the gap between their Houses?
I am going to go with no. Gryffindor and Slytherin already seemed to be at odds, without Harry or Draco's help, from the previous years. Slytherin seemed to have been alienated from everyone already. If you take the line that Ronald said, and, assume that that is how the rest of the wizarding world feels as well, nothing short of Harry defeating Voldemort that very second, as a member of Slytherin house, was going to bridge that gap in Year 1.