This is a question of interest to most mathematicians who are research active and not slowly but surely knocking off important problems in their field at the rate of one per paper. (I think I could have ended the previous sentence at the word "active" without much affecting the meaning!)
I think the answer is ultimately quite personal: you are free to set your own standards as to how much of your work to publish. I myself understand the psychology both ways: on the one hand, math is usually long, hard work and when you finish off something, you want to record that accomplishment and receive some kind of "credit" for it. On the other hand, we want to display the best of what we have done, not the entirety. This position is well understood in the artistic and literary world: e.g. some authors spend years on works that they deem not ready to be released. Sometimes they literally destroy or throw away their work, and when they don't, their executors are faced with difficult ethical issues. (This is roaming off-topic, but I highly recommend Milan Kundera's book-length essay Testaments Betrayed, especially the part where he details the history of how after Kafka's death, his close friend Max Brod disobeyed Kafka's instructions and published a large amount of work that Kafka had specifically requested be destroyed. If Brod had done what he was told to do, the greater part of Kafka's Oeuvres -- e.g. The Trial, The Castle, Amerika -- would simply not exist to us. What does Kundera think of Brod's decision? He condemns it in the strongest possible terms!)
Another consideration is that publication of work is an effort in and of itself, to the extent that I would not say that anyone has a duty to do so, even after releasing it in some preprint form, as on the arxiv. A substandard work can be especially hard to publish in a "reasonable" journal. I have a friend who wrote a short note outlining the beginnings of a possible approach to a famous conjecture. She has high standards as to which journals are "reasonable", and rather than compromise much on this she determinedly resubmitted her paper time after time. And it worked -- eventually it got published somewhere pretty good, but I think she had four rejections first. I myself would probably not have the fortitude to resubmit a paper time after time to journals of roughly similar quality.
As you say, though, one advantage of formal publication is that the paper gets formal refereeing. Of course, the quality of this varies among journals, editors, referees and fields, but speaking as a number theorist / arithmetic geometer, most of my papers have gotten quite close readings (and required some revisions), to the extent that I have gained significant confidence in my work by going through this process. I have one paper -- my best paper, in fact! -- which I have rather mysteriously been unable to publish. It is nevertheless one of my most widely cited works, including by me (I have had little trouble publishing other, lesser papers which build on it), and it is a minor but nagging worry that a lot of people are using this work which has never received a referee's imprimatur. I will try again some day, but like I said, the battle takes something out of you.
Finally, you ask about how it looks for your career, which is a perfectly reasonable question to ask. I think young mathematicians might get the wrong idea: informal mathematical culture spends a lot of time sniping at people who publish "too many papers", especially those which seem similar to each other or are of uneven quality. Some wag (Rota?) once said that every mathematician judges herself by her best paper and judges every other mathematician by dividing his worst paper by the total number of papers he has published. But of course this is silly: we say this at dinner and over drinks, for whatever reasons (I think sour grapes must be a large part of it), but I have heard much, much less of this kind of talk when it comes to hiring and promotion discussions. On the contrary, very good mathematicians who have too few papers often get in a bit of trouble. As long as you are not "self plagiarizing" -- i.e., publishing the same results over and over again without admission -- I say that keeping an eye on the Least Publishable Unit is reasonable. Note that most journals also like shorter papers and sometimes themselves recommend splitting of content.
So, in summary, please do what you want! In your case, I see that you have on the order of ten other papers, so one more short paper which is in content not up there with your best work (I am going entirely on your description; I don't know enough about your area to judge the quality and haven't tried) is probably not going to make a big difference in your career. But it's not going to hurt it either: don't worry about that. So if in your heart you want this work to be published, go for it. If you can live without it, try that for a while and see how you feel later.
I did exactly as others have suggested: created a fake email account (this was pre-Gmail; I think I used "sithmail") called [author name]reviewer, and corresponded that way.
Actually, I did this not to ask about the paper, but to strongly recommend to the author that he put his article on the arXiv before signing anything (and pointed out that the editor himself frequently posted to the arXiv). I didn't see any point in putting the editor into a difficult position, about whether he was allowed to suggest to the author that the author arXiv the article.
Best Answer
The answer to your question really depends on how much work required justifying their claim. In my personal experience:
Therefore if the justification did not require much work, I would give it to the authors of the paper and let them publish it. Then you would be mentioned as an anonymous referee.
May I submit the result as my own? If so, must I wait until their revision before I submit my own paper?
That I do not really understand. Since they do not have justification of the claim, they cannot publish it. If they come up with a justification, your proof is of not much value since they already have a proof. In my opinion, you must not publish your proof before their paper is published.
If proving the claim is really difficult and required a lot of work, I do not really know what to advise. In one related situation I did as follows:
I was a referee of a paper $X$. However, before refereeing the paper I already knew how to prove a much better result (5 pages of elementary calculations vs 40 pages of a difficult proofs of far better results). But I only had my proof in a draft. I wrote to the author that I was a referee and I suggested that he would withdraw the paper from the journal and we would publish a joint paper. This is exactly what happened.