I think your question is so important as to deserve multiple answers, even if there is a good deal of overlap among them. (Indeed, overlap indicates that various respondents feel the same way about something, which is important.) So:
1) Should you summarize the main results and or the argument? If so, how much is a good amount? What purpose does this serve?
Absolutely yes. How much is hard to say: one thing I often do is try to put the paper down and write out a summary of the results (including precise statements) and the basic strategy of the proofs. This gives me something very longwinded. Then I try to pare it down until the point where cutting out anything else would be an obvious disservice to the author or to my explanation. It serves several purposes:
a) You need to have a high level grasp of what is being done in the paper. Writing out this summary allows you to gain this grasp (and this is especially important if the authors have done a bad job of indicating the importance of the paper in the introduction; I find that perfectly good papers are often not properly explained in this way) yourself,
b) You demonstrate to the author and editor that you have actually read, absorbed and digested the paper. (Or you demonstrate that you haven't!) This is important both as part of the decision process and also psychologically for the author: it is really maddening to get a paper rejected by a referee who you think didn't properly read it or understand it.
c) This factual information forms the basis for the opinions you will render later on.
2) What do you do when the journal requests you to evaluate the quality of the result (or its appropriateness for the journal)?
Then you evaluate the quality of the result and its appropriateness for the journal! This is the most important part of the refereeing process, and you should do it even if you aren't explicitly asked to. (It is very unlikely that the additional opinions you provide will be unwelcome. It is much more likely that you just didn't get sufficiently explicit instructions for whatever reason.)
Should one always make a recommendation regarding publication?
Yes. It needn't always be "absolutely yes" or "absolutely no" -- i.e., you can recommend publication conditional on the author's making certain changes (possibly without looking at the paper again, I mean) or you can recommend publication more or less strongly according to certain information that the editor has and you don't (e.g. backlog), but you should definitely make a specific and unambiguous recommendation.
3) What to do about papers with major grammatical or syntactical errors, but that are otherwise correct? Does it matter if the author is clearly not a native English speaker?
This is a tough one. I think if the errors are truly grammatical and/or syntactic and they are not so serious that they create ambiguity in the meaning or interfere with your undertanding of the paper, then this should be regarded as a "copyediting" issue and be clearly placed at a lower level of importance than matters of mathematical correctness or worth.
As an author, it can be annoying to have the recommendation for publication delayed by a referee who wants to make a big fuss about copyediting issues. (Even more annoying are referees who delve into issues of formatting. I have had referees ask me why certain things are italicized and one referee who sent me an entirely new copy of my paper with the font size changed. I felt that was silly.) By coincidence I got a referee report just this evening, and -- while it did point out a matter of content, for which I am quite grateful -- there were a lot of nagging little things, e.g. "Your notation for cyclic group seems odd to me." Okay, thanks for sharing.
Ideally, there will be an actual copyeditor who steps in after the paper is accepted. Some journals do this (mostly AMS and MAA journals, if memory serves).
4) On this note, at what point does one correct such mistakes? Ever? If there are fewer than a dozen? Should one be proof-reading the paper?
That's totally up to the discretion of the referee. You certainly have the right to do it but are not obliged to. I have myself done it sometimes for authors who are both junior and non-native speakers of English, because I think that they probably will both be happy to receive it and learn from it.
5) What do you do when you do not understand an argument?
Well, that's your biggest worry when you're refereeing a paper, isn't it? I can't give a complete answer here. Certainly you should be prepared to put a fair amount of effort into understanding a crucial point in the paper, including asking friends and colleagues about background knowledge, if appropriate. If after a while you still don't get it, you should say so in the report, and in my opinion you should make clear to what extent the author is at fault for the lack of understanding and what you want to be done about it. As an author, I have gotten comments from referees like "I don't understand the proof of Theorem 13." What am I supposed to do with that information? Tell me exactly what you didn't understand, or tell me that the whole argument is hopelessly vague or flawed, or tell me that you were honestly in a little over your head at this point. But tell me something.
Does it matter if it "feels" correct?
Yes, I think it does, but I don't feel comfortable saying more.
How long should one spend trying to understand an argument?
Again, there is no one good answer to this.
6) What to do about papers that have no errors but whose exposition is hard to follow?
I would say that if you can successfully ascertain that there are no errors, the exposition was not that hard to follow! Seriously, you can make as many specific suggestions/demands for expository changes as you like, but if you have actually understood and verified the paper, then rejecting it just because it is badly written is not a decision I would feel comfortable making as a referee. (As an editor, yes.)