Welcome to my (very) long answer of answers. I hope that these give you an indication of where I stand as it pertains to moderating this site, and also gives you a better idea of who I am. Cheers!
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?
Such a circumstance is something that I feel that moderating team has to strive to achieve consensus among themselves, as moderators can undo even the quite serious actions of other moderators. I do feel that no single user is greater than math.SE as a whole, and that quality answers will continue to be submitted even if any one individual member leaves; we need look no further than the unfortunate departure of Arturo Magidin as evidence of this. Truly problematic users may also inhibit other equally qualified people for joining our site as contributors, and this opportunity cost may be easy to overlook. It is only right that the moderators follow in broad outline the guidelines set out in this post by Jeff Atwood, meaning that there would be a series of responses of increasing severity. Whether the increments would be exactly the same would have to be decided on a case-by-case basis, but the moderators do neither the math.SE community, nor themselves, a favour by giving highly problematic users significantly more leeway based on other contributions.
How would you handle a situation where another mod closed/deleted/etc a question that you feel shouldn't have been?
Much of this depends on the actual circumstances of the closure/deletion. I'll restrict my attention to the closure case, as the deletion case is similar, just with different numbers and no current meta-thread for its undoing. Should a moderator vote to close a question as the fifth of five, I would have absolutely no problems with this; the moderators are users, and are thus as entitled to express their opinions on valid/invalid content on the site as any other (sufficiently privigeled) user. Seeing a moderator vote as the fifth of five would be an indication to me that restraint was shown, and I should only reciprocate. Depending on how strong my feelings are about the particular instance, I would have no problem adding another answer to our Requests for Reopen Votes meta-thread, and hope to become the fifth of five in the opposite direction.
If, on the other hand, another moderator has used their sledgehammer in a manner I feel was inappropriate, the worst thing I could do is return in kind. My first step would be to contact that moderator and try to get their thoughts on the matter; it is perfectly possible that they saw or knew something about the particular case that I was unaware of. If an agreement still is not found after these discussions, I would again hope to guide the community in having this action undone.
Have you ever been suspended and if so for what? Would you allow existing moderators to check the veracity of your answer?
No, I have never been suspended. I have never been warned by the moderators. I'm doubtful that a moderator has ever found a flag on any of my contributions "helpful" (though I cannot be certain of this). I would absolutely allow the moderators to check and report on ther veracity of this response. (Even more, I'm kind of curious what sort of notes (if any) the moderators have on me.)
How will you handle situations where the community consensus contradicts your personal opinion on specific moderation policies?
I strongly feel that moderators should not use their powers in activist ways. At the same time there are limits to what I would do with these sledgehammers in the name of "community consensus". For example, I would not suspend, for example, Nazem Kadri simply because there is a demand by the community for this to be done. While this might seem like an outlandish possibility, I think it fits within the broad parameters of this question.
I do understand that the impetus of this question is likely what have become known as PSQs, and the possibility of there being a community consensus to close/delete them, and so I will address this more directly. First of all, I have no idea what form "community consensus" will take with respect to this topic; even though recent meta-threads have indicated that there is support for such policies, I do not think anything close to a consensus has been achieved (and the fact that conversations continue makes me certain that I am not alone in this thinking). Even more, I have not seen any proposals to the effect that "PSQs should be flagged and duly closed/deleted by the moderators". I hope that we never see such a proposal; and in general I would be discouraged by any proposal that obliged the moderators to use their sledgehammers. The actions of the individual users will go much further than the discussions on meta in shaping community norms (if not policies). I will not unilaterally undo the actions of the community as it regards the closing/opening/etc of questions, but I do not (and cannot) promise to never start a discussion about having the community itself undo these actions on a case-to-case basis. (I don't expect this to be a common occurence, but it would be somewhat evasive of me to not admit the possibility.)
What will be your policy on questions from various ongoing contests from math journals, contests which are not proctored such as the USAMTS(http://www.usamts.org/) and questions from national olympiads which are part of the process of selection of national teams for contests like the International Mathematical Olympiad posted against the rules of the contest on this particular website (there have been instances where questions of some mathematical contests have been posted before the date) ?
I strongly believe that math.SE is part of the larger mathematics community, and as such we should strive to be responsible actors within it. A large part of this is to not hamper the efforts of other facets of this community. I do feel that it is more than appropriate to temporarily close and delete all questions for which there is strong evidence to support that they are part of an ongoing contest or exam. This is one of the very few instances I believe that moderators are justified in using their sledgehammers. But once deadlines have passed, I see no reason why such a question should remain deleted and closed (and here, again, a moderator sledgehammer is useful).
I also do not feel it is the job of the moderators to hunt down contest/exam questions, or to treat questions which only might be from contests/exams as if they were certain infractions. I would hope that math.SE becomes known as a responsible part of the larger mathematics community that professors and contest organisers feel they can monitor this site, flag posts when cheating is detected, and see a quick response. I recall that an organiser of the Canada/USA Mathcamp had personal contact with Qiaochu Yuan, and this lead to several questions being closed/deleted. Unfortunately, I also believe that this contact was only through Qiaochu, and I hope that a relationship of this kind can be established with the current (and future) moderating team.
This is a question to those candidates who do not have the Deputy badge (which is given for 80 helpful flags): Dealing with flags is a big part of moderation. How do you think that your lack of experience is going to affect your handling of the flags? And how do you expect to compensate for it (at first, of course)?
Well, I certainly have not achieved the Deputy badge. I'm actually still quite a long ways off. I don't know how to explain this, except for possibly two contributing factors. First of all, the tags that I have spent most of my time in do not seem to attract too many contentious users (some, but not too many), and so I rarely come across posts/comments that I deem flag worthy. Secondly, I probably have a more laissez faire attitute toward contributions to the site, looking for ways to accept rather than flag them. Even those times when I try to hunt for flags I find only very few (and the Review queue seems to empty extremely quickly). (As a note, of the 50 posts I have flagged (as of this writing) on the main site, all but three were deemed "helpful" and the three stragglers were "disputed" as opposed to "declined".)
If my +2 Cap of Precognition were not in the dry-cleaners, I could better answer how I will compensate for this "lack of experience". One thing that will be quickly noticed by myself is the sort of posts/comments that are flagged, but I also plan to become acquainted with the manner in which the current moderators are handling flags. I would imagine that for a time some users may see more of their flags being declined, but I would not expect this to last long as I become conditioned to the flag-handling norms. (This entire answer is likely a great example of "How Not to Be Elected". But it is truthful. Except for the having a +2 Cap of Precognition part.)
How involved are you with the meta site? Do you expect this to change if you are elected? Are you participating in most of hot threads? Are you just reading them and participate from time to time? Are you reading meta threads regularly without participating? Do you even check the meta site?
I feel that I am at least moderately involved in meta. My active involvement ebbs and flows depending on exactly what is going on around the site, and exactly what sort of questions are being asked. I am often amazed at the knowledge that certain users have with the workings of the SE framework, and other technical matters. But I certainly keep abreast of what is happening, checking meta quite religiously (or is that fanatically?), and feel I know a fair bit about the habits and personalities of the usual meta denizens.
It did take me a while to find my voice on meta, as it can be a daunting task to follow the various strong personalities and histories in the more contentious threads. I hope that when I do participate (and I am speaking about my current non-moderator instantiation) it is seen as a more reasoned and moderate voice. And I fully expect this to continue in any future instantiation.
What, in your opinion, is the purpose of closing, as opposed to protection, locking, and deletion?
The closing of questions has a dual purpose. Perhaps at is basic level it is an indication to the OP that there is something wrong with the question as it currently stands, but there is still a possibility to correct it, and have it re-opened. Secondly, it is often the first step towards other more serious actions (such as deleting).
Deleting questions is generally reserved for those cases where the question likely has no chance of being redeemed, and it is better for it to simple not exist to the vast majority of users. Of course, deleted questions actually do exist, and deletions can be undone, making it a very valuable method for hiding content in cases as discussed in an above answer.
The locking of posts is usually done when there has been an inordinant amount of activity (e.g., continual edits and reverts), but it can also be applied to cases where discussions in the comments have gone way off course and have become inflammatory, thereby giving the moderators time to separate the wheat from the chaff before unlocking with a (hopefully) more civil continuation. I'll admit to not understanding how protecting questions actually works in practice. I understand that it is for those cases where many non-answers by new users have either been submitted or are expected, but the very low reputation minimum to post a new answer makes me think that its purpose is to simply scare users into not providing new answers.
One of my basic beliefs regarding these methods is that, when possible, in the vast majority of cases they are better done by the community at large. I have seen questions closed rather quickly without moderator intervention, and for the most part any "community consensus" has to start with the actions of the community shaping a community norm. If moderators are to be "human exception handlers" then it stands to reason that the events they handle should be exceptional.
Suppose you were to disagree with the other mods on the closing / deletion of a post; what action(s) would you take?
(I think this was answered above...)
Have you had any serious or acrimonious dispute with any other Math.SE user? How do you feel about the outcome?
The worst "disputes" I have had with specific math.SE users haven't gone beyond simple frustration. There was a very real possibility of my entering into a bad dispute with a user over one comment that I initially took (and to a certain extent still take) as a personal insult. I submitted a poorly thought out, somewhat vicious comment in reply. I deleted this fairly quickly, and replaced it by a comment to the effect of "I won't deal with you again, but here's why you're wrong." I think I almost handled this well, but wish that initial comment was never submitted. (For the relevant post, see here.)