There is a lot of noise in voting, so don't overanalyze it. There is a correlation between good comments and upvotes, but unless you get at least -3 on a comment, or perhaps -1 on 5 comments in a row, you should probably just ignore it. Also, upvotes usually mean you did something right, but of course a comment made early in the debate gets more visibility and votes than a comment made late in the debate.

Generally, upvotes and downvotes mean "want more of this" and "want less of this". This is a community about rationality, so you should consider whether the given way of communicating contributes to rationality, or more specifically to building a good rationalist community. Use your own judgement.

Going against the majority opinion... I'd guess it depends on whether the argument brings something new to the discussion. Saying: "you are all wrong because you didn't consider X" (where X is something that makes sense and really wasn't mentioned on LW) will probably be welcome; saying "you are all wrong, because this is against my beliefs / against majority opinion" will not. But here I would expect even more noise than usual.

By the time I get to wreckage of a flame war to catch the juicy details, there isn't much point to myself as an individual actor dishing out further downvotes.

I'm not confident with how to proceed in upvoting posts that already have lots of karma.

I think you should upvote or downvote comments regardless of the karma they already have. If 10 times more people think a comment is awesome (or horrible), it should get 10 times more upvotes (or downvotes). If you worry about someone getting infinitely many downvotes for participating in one stupid flamewar -- they have an option to retract the comments, which stops the voting.

The intuition behind this is approximately that if voting is better than not voting (which based on my experience with web fora I consider obvious) then more voting is better than less voting. Also, some fraction of users will abuse the voting system, so we need more votes from the nice users to balance this. Actually, this is also a reason why the nice users should vote more often.

I'm sometimes worried about my votes being biased in the sense that they go to posts, or comments, which increase the visibility of things I only value personally, rather than being reflective of how much a given post, or comment, increases, or decreases, the quality of the discourse on Less Wrong.

You are a human and to some degree this is inevitable. You should try to do as well as you can, but don't try to reverse stupidity and extract signal from noise. Let's assume that 20% of your votes are biased, but 80% correctly estimate what improves the discourse. How could you improve this ratio by voting less? You can't; because the assumption was that you don't know which votes belong to the 20%. Voting less frequently is equivalent to giving your votes a multiplier 0.5 or 0.1 or maybe 0.01. If you have a reason to believe that you are significantly more biased than an average voter (which is not the same as an average commenter), you should abstain from voting completely; otherwise there is no reason to tune down your voting. The mere fact that you care about this all is an evidence that you should vote.