On (1), the mental model I have is more like an understanding of who the super-intense supporters are. 90% of people might say they support background checks for gun purchases, but it's the 10% who support it who are voting on the basis of that issue, in part due to organizations like the NRA. I think our findings are consistent with that, in the sense that politicians seem to believe they don't need to learn what public opinion is in order to get re-elected; they can focus on other forms of information-gathering.
On (2), I suspect there is something there, but it's hard to make sense of Democrats from that angle, as Democratic primary voters, for example, are more liberal than the average person, yet Democratic politicians don't seem to see their districts as more liberal than they are. My guess is the key group is something more like "the people who choose to write in and go to meetings."