In the US. the President is voted upon separately, and often is the opposite of the party that runs the House of Reps. Party footprint in the House has little baring which party in the White House.
The party whip rules with an iron fist - votes where it is safe for party members to vote their conscience or directly constituents are the exception, not the rule. And my opinion is that a big reason for this is that individual representatives are basically anonymous here in Canada. They're too small to get any media coverage. The only time an MP or MP candidate makes the news is if they have a Cabinet position or if they make an apocalyptic gaffe. Debates, policy, etc. are ignored below the leadership.
No, there wouldn't. The electoral forces that drive duopoly operate the same almost independently of district size, changing the number of single-member House districts with first-past-the-post elections won't reduce duopoly.
Changing the electoral method might (and doesn't require changing the number of districts.)