How would you deal with a user who produced a steady stream of valuable answers, but tends to generate a large number of arguments/flags from comments?

Working on the assumption that looking at the incidents confirm the user in question is at least to a large part responsible for the arguments, and the arguments are not only with a few particular users: We have the overall rule to be nice. If somebody habitually violates that rule, that is not acceptable. Nevertheless, except in the severest cases, one should check one's own judgment by asking for a second opinion, so consultation of fellow moderators before any disciplinary action is the default. The first step would be an unofficial request to moderate the behaviour/language. Many people do calm down when asked to do so politely. In severe cases, that stage could be skipped. Afterwards, if the situation doesn't improve, slowly escalate. The next stage would be a warning or moderator message (I don't know if those are different things, and if so, which is more grave, so this might be two stages). If necessary, suspension would be the next step, in unfortunate cases, repeated suspensions.

If the arguments are with only a few particular other users, it is likely a case of "it takes two". In that case, requests for moderation would be sent to all parties. The option to try and just ignore each other should probably be mentioned before sterner measures are taken. The following like above, where as long as no party moderates their behaviour, warnings/suspensions would be issued to all parties.

How would you handle a situation where another mod closed/deleted/etc a question that you feel shouldn't have been?

Discuss the case with the other moderator, laying down my reasons why I object to the action, and asking for the reasons that led to the action. If that does not resolve the dissent, involve the other moderators in the discussion. If that doesn't lead to a resolution - not necessarily complete consensus, a clear majority of opinion should suffice - it is probably the time to let the community discuss the matter here on meta (or, if confidential matters are involved, with the community managers).

In your opinion, what do moderators do?

In short: mopping up in aisle seven, and cooling down disputes.

Judging by these data points, a large part of moderation is handling flags, as expected. I have no data about what sort of flags makes up which proportion of the flags, so I can't guess whether investigating suspicious activities (sockpuppet accounts, suspected voting rings, other things I can't at the moment think of) would be a rare or a not-so-rare duty. Hopefully rare. Part of flag-handling involves deleting posts-that-should-not-be, nonsense like asdfasdf posted as answers, questions posted as answers to other questions, troll posts, if those are not deleted from the review queues or by 20k users in a timely fashion; sometimes converting answers to comments. More serious are offensive posts, which ought to be deleted on sight, without waiting until enough users with the power to vote for deletion have seen them.

The most delicate part in flag-handling - except for the detective work about forbidden behaviour mentioned above - I can think of now is dealing with flags that arise from a conflict between users. There, a moderator's part is pacifying and trying to guide the conflicting parties to a compromise, to moderate.

Spam, I have learned, should be flagged as spam also by moderators, since deletion by spam-flags produces data for whatever anti-spam software is employed, which moderator deletions don't.

Also a large part (larger than expected by me) is apparently cleaning up comment threads, deleting obsolete, unconstructive or offensive comments, culling excessively long comment threads.

Then there are things also experienced users that aren't moderators do: editing posts in need of editing, explaining the workings of the site to newcomers, reviewing.

A diamond will be attached to everything you say and have said in the past, including questions, answers and comments. Everything you will do will be seen under a different light. How do you feel about that?

Slightly uneasy, to be honest. But it's not much of a change. I already have my real name next to my posts and comments, and I prefer that to be associated to useful contents rather than inappropriate remarks.

How would you use the powers conferred to moderators to be more effective than just the powers earned by standard users for hitting 10,000 or 20,000 reputation?

Concerning the things ordinary users can do too, but moderators can do more:

Flags would direct me to problematic posts that others have discovered, and not only those that I happen to come across while browsing the site. The one-click-deletion powers of a moderator would allow me to remove posts that really shouldn't have been made quicker than flagging as not-an-answer or low-quality, voting to delete, and waiting for enough other users to cast their votes too.

And a "please be civil, people" from somebody with the power to suspend carries more weight than one from ordinary users.

Then there are the things ordinary users can only flag and ask moderators to do, like merging duplicates or migrating questions (not without consulting the moderators of the prospective target site). And suspending users, which hopefully isn't a too frequent occurrence.

Have you ever been suspended? If so, provide particulars, including when, why, and for how long.

Not to my knowledge. A moderator is welcome to check for holes in my knowledge of that topic.

Since the meta site is a completely different website, where policy issues are handled, how versed are you in the meta site? If you're not posting many questions or answers there, do you at least follow it passively and read the discussions and the comments there?

I'm reading it regularly, so I have a rough idea about the current affairs. Occasionally, I can even answer a support/information request question there before somebody else does. From the history, I have only read parts. Though the traffic is low enough to follow current affairs, meta is too large to read the record completely.

What is the most serious problem facing Mathematics Stack Exchange today? And what would you hope to be able to do about that problem as a moderator?

I think Ilmari Karonen nailed it, the biggest problem is size/growth. With currently between 600 and 700 questions per day, the site has grown too big for any single person to keep an overview of the entire site. The sub-communities will become more separated from each other, and will develop different strategies to deal with the increasing question volume and in particular the low-quality questions. The transition from the village where everybody knew everybody to the big city where every quarter has its own customs will not be painless.

Extrapolating from what I saw on Stack Overflow, seeing that my neighbour does things differently from me can cause a lot of friction. Suppose in tag x, the consensus is that Problem Statement Questions are unacceptable and need to be closed put on hold prontissimo as well as downvoted, while in tag y the users consider PSQs acceptable. Imagine the hilarity when a PSQ is tagged with both tags. I've seen such occurrences lead to very hostile comment exchanges.

What can a moderator do about that? Quenching the fire when conflicts arise.

How would you personally prefer that so-called PSQs [Problem Statement Questions] be handled? Given the current policies and customs of the site, how do you think that they should in fact be handled?

Like others, I consider PSQs undesirable.

I would prefer that PSQs are edited to become non-PSQs before they are answered. If they are edited to become adequate questions after they have been answered, that is still okay. But a lot of them never are edited, and I believe that having a load of verbatim assignments including the exhortation to "show your work" with answers on the site sends the message that it is okay to ask such questions here. I don't like that message. I think context (I tried this, got stuck there) makes the difference between "do my homework" and "can someone help me do my homework". I like the latter, but not the former. Therefore I think such questions should be put on hold until edited, and then reopened.

But as long as there is no clear consensus on the treatment of PSQs (and I don't expect one site-wide), I won't single-handedly put them on hold as such. I would have no qualms to single-handedly mark them as duplicates if they are.

This is aimed to the candidates which do not have the Deputy badge, and no reviewer badges for the queues available to them. How do you think you'll handle flags, when you've yourself flagged successfully only a few times?

I have the Deputy badge, so a bit of experience with that side. Handling flags is a different matter, however, I have no experience with that yet.

Do you expect being a moderator to affect your involvement in other aspects of the site (e.g., review queues, editing, posting questions/answers)? If so, in what ways? If not, why not?

I expect my answering rate will drop considerably if I am elected, since moderation duties will take time. And my comment rate. Whether the time spent reviewing and editing will increase or decrease depends on how much time other moderator duties take.

What is your opinion of efforts to delete questions that have correct answers? Does it matter whether they are votes to delete or downvoting to help the autodelete process apply to the question?

If the question is bad, merely having correct answers is in my opinion not sufficient to prevent deletion. The answer would need something that raises it above being merely correct. I have cast delete votes on questions having correct answers. Better answers have stopped me from voting to delete questions I would otherwise have been happy to see gone. One judges whether the gain from getting rid of the bad question outweighs the loss of the associated deletion of answers in the "Delete votes" queue (/tools?tab=Delete for 10k+ users) and casts or withholds the vote accordingly. Reasonable people can come to differing conclusions in such matters, so some posts leave that queue without being deleted, others get deleted and later undeleted, yet others get deleted and stay deleted. Since moderator deletions cannot be undone by ordinary users, being elected would end my casting delete votes there, it's up to the part of the community with sufficient privileges to decide in such cases, and the community must have the power to reverse such deletions on its own.

With regard to downvoting and the autodelete process, I don't think that is a matter moderators should be concerned with. Voting is everybody's own decision, and as long as it's not targeted at specific users, should remain private. While I don't like the practice, I have no big problem with it either.

As a moderator, you will often find yourself interfacing with upset users, resolving conflicts, or issuing suspensions for out of line behavior. These situations require empathy and emotional intelligence. Do you practice compassion in your non-mathematical life? In what ways have you prepared to facilitate a harmonious community?

I do have difficulties imagining how other people feel, so that's a minus on empathy and emotional intelligence. But roughly fifty years of life have taught me to be patient and to stay calm when things heat up, or if I don't stay calm, to step back and calm down again. Mostly.

A further minus on preparation, I have no previous experience of moderating or leading a community.

But I am willing to hear out both sides of a dispute before forming my judgment, so I may be up to the task nevertheless.