I think many of us here are the same way (or were at some point or sometimes still are). I love to argue. Often that comes from a good place, but often it comes from just that desire to win or to be difficult or whatever. I still have to push myself to really listen to the other view. Then my first step (when I'm in a better place) is to literally ask myself something like these questions, in roughly this order:
+ What does this person believe?
+ Why does this person believe those things?
+ What would I need to believe (first) in order to convince me of those things?
Try to short-circuit evaluation: don't judge the thoughts at all. Don't ask if you agree. Just figure out what they believe and why they might hold those beliefs. What other beliefs seem to be required? What other beliefs would make someone think that way? This already gets you fairly far into their mindset.
Also like your last bit of advice - think about the person's point of view before evaluating what they've said. And by doing this, I see how one could also get to the root of a disagreement, rather than just the surface, and address that instead. Everything becomes philosophical from there I'm guessing.
+ What does this person believe?
+ Why does this person believe those things?
+ What would I need to believe (first) in order to convince me of those things?
Try to short-circuit evaluation: don't judge the thoughts at all. Don't ask if you agree. Just figure out what they believe and why they might hold those beliefs. What other beliefs seem to be required? What other beliefs would make someone think that way? This already gets you fairly far into their mindset.