That's not how it works. The Congress as a whole does not represent the US population as a whole; it's not supposed to, and I'd argue it would be a terrible idea if it did. You have one representative, who represents one district. If the whole Congress represents the whole country, then no individual representative represents any specific area.
If a district in Vermont votes 60% R/40% D in favor of a Republican, and a district in West Virginia votes 5% R/95% D in favor of a Democrat, then that argument would say since the total of the two districts is 32.5% R/67.5% D (assuming equal population districts), the "congress" of two districts should have two Democrats, rather than one of each. The Vermont district should have no effect on the representation of the West Virginia district.
Frankly, I would consider myself more towards the "anti-democratic" side: I'm much more on the side of "democracy is two wolves and a sheep voting on what to eat for dinner". Not that the foundation of government shouldn't be democratic, but there need to be very solid protections in place to prevent a majority trampling on a minority. If anything, our system has become too democratic recently; very few of the rights in place to protect us are still respected. (I guess our 3rd Amendment rights are still pretty safe, but other than that.)
I know how it works. The Congress that we get reflects the gerrymandering of Congressional districts that has been done state by state. The average of people elected to Congress is not biased Republican because voters in Vermont and West Virginia are not balancing out, but because states like Florida and Ohio have districts drawn such that each district is clearly one side or another, with the number of Republican districts significantly exceeding their popular support across the state.
The national result of those gerrymanders is to give Republicans a non-representative edge, and to give the extremists of both parties a much bigger impact than they would otherwise have. And this is not just "how it has always been". This is a trend that is intensifying over time. That is, Congress does not represent the people very well, and represents us less well now than it did a generation ago.
PS You picked a poor example. The only court case in which the third amendment was at direct issue is http://en.wikipedia.org/wiki/Engblom_v._Carey in which it was found that the law was so obscure that bureaucrats could not be faulted for having violated it. Any future violation can also be defended on the same grounds, so it is effectively null and void.
I believe that there are two, related problems in the American democratic process. The most basic one is the power of incumbency and lack of competitive elections.
Turnover is at a historic low and getting worse. It's lower than in the Soviet Politburo, and most districts never have competitive elections. The last mid term was at least partially a reaction to that. However en-mass replacement is no better than perpetual incumbency, and blindly voting against everything is no better than being a rubber stamp. I think they're equally bad. To those that say it's worse - would it really have been bad if there had been congressional opposition to the Patriot Act and Iraq war from someone other than Bernie Sanders and Ron Paul?
To get better people, there needs to be rotation in office in a regularized fashion. Yes, that means Term Limits. I think limits on consecutive terms is a better than absolute limits. I.E. you can't run for the same seat in the election immediately following 2 consecutive terms, but can in the election immediately following. This solves the problem of incumbency without permanently barring good people. It's the system that was used in Greece and Rome, and was favored by many of the founding fathers.
It's even more helpful considering that currently, power in Congress and Senate is essentially based entirely on seniority.
Regarding Democracy vs. Law/Rights:
How do you suggest law and rights be strengthened? The staleness of congress is just leading to more and more use of executive power, more arbitrary and king-like rule.
If you want laws that are better and more followed, you need better people to write them, which is congress.
Yes, representatives represent districts, not the population as a whole. But the fact that a majority of voters voted for a Democratic representative for their district, yet the House as a whole has a solid majority of Republicans, indicates that the districts are highly gerrymandered rather than being apportioned in a way that reasonably reflects the population. And if you look at the maps of district boundaries, you find that yes, they are highly gerrymandered.