-Are there noticeable differences in how the majority of men and women communicate contentious ideas (like politics, sex, and religion
?
Although this may be a sexist statement, men generally try to keep it factual and logical and attempt to keep emotions out of it, whereas women may be more likely to put emotion first. Personally, I think men or women who fight tooth and nail based on facts and stats which are easily twisted towards any direction are just expressing their emotions in a different, more socially acceptable way.
-Do you ever find yourself getting so caught up in what you have to say that you forget to listen (I know I do)?
I think we all like to think we listen, but when we feel strongly about something or skim through and/or ignore posts that follow an OP. But here again, I think the ladies are more apt to be open-minded. One like me, who's not as educated as some/many of you, would tend to listen to what everyone's got to say before they form a strong opinion about something. I think things are changing, but when I was growing up, the men talked politics while the women talked housekeeping.
I believe this point is crucial. Often times, we catch an OP and skim or choose one aspect of the statement or question to respond, ignoring the intention of the post to put forth our own platform. Becoming a more familial and tolerant society, both at TFP and in the world, would suggest we need to listen to one another, not pick and choose what we want to hear.
-Is it more important to prove you are right, or to try see the other side?
I love to see the other side. If it makes sense, I'm easily won over. Maybe a competitive nature would make some feel like they "lost" if they admitted a different opinion had some validity?
-Can our positions' evolution be improved through opposing points of view?
Definitely! A great debate can lead to discovery, which opens minds and hearts. Heck, it could lead to world peace. 
-Do you think your positions CAN evolve, or do you not WANT them to?
Yes! I love to grow and learn and understand where others are coming from. If someone can argue a point while listening to others' opinions without imposing condescending tones, I think even the superior politicos can learn something.