I think that this is a great idea. Here are a few thoughts I have:
Structure of Challenges
I personally do not think that there is a shortage of good answers on this site, but rather good questions. Consequently, if we had to choose, I would recommend centering the challenges on excellent questions; I'm certain that excellent answers will follow since they come with their own rewards. However, I'm only saying this from a logistic point of view. If it's quite feasible to recognize both best questions and best answers, it would certainly be better.
A week should be sufficient for each round, and will encourage quality with quantity. Only getting a few good posts every couple of weeks is not much progress, but if we can encourage a bunch of great questions and answers every week, we'll increase both the quality and the activity of the site.
Just a little suggestion from my experience at Ludum Dare: every once in a while, it may be refreshing to have a volunteer user "host" a challenge, meaning that they choose the topic (perhaps more specific and related to their tastes) and make the announcement themselves. It may add some extra flavor to the challenges.
Promotion and Presentation
A system message is definitely crucial for getting people informed. I would also recommend a meta post prior to each challenge, with the following information:
- Topic of the challenge
- A brief explanation of the topic; a definition, perhaps an example question from the past, and maybe a few informative links, just to get people started
- Maybe an inspirational quote from a philosopher, related to the topic to help people think
- Contest details (or a link to a post with the details)
Obviously, getting people interested is the biggest challenge, so (as an extension of this header):
Although reputation is a great way to motivate people, there are still only a limited number of people who can/will participate in the challenges. Unless we can get more active users into the site, the winners will ultimately be from the same group of a couple dozen people (this is an observation from my short time on this site), and this will discourage most people from getting involved. If we want to promote participation in the site, perhaps we could recognize winners in two categories: highest voted question/answer, and highest voted question/answer by a new (<200 rep?) user (I realize this is a bit flawed, I'm just suggesting off the top of my head).
This way, everybody will be encouraged to get involved and contribute great questions, without being intimidated by the sight of high-rep users always winning. This will also provide great models for new users; they can take inspiration and try to replicate the quality of past winners. Even if they don't get the highest score overall, they may still get recognition for best question from a new user, which is definitely a confidence-builder. If they do a good enough job, maybe they'll win both categories!
This can act as a sort of pipeline, directing new users through the less challenging category and leading them, with encouragement, to participate actively in the high-level category.
Given the significance the topics have for philosophy, each challenge will serve to increase our search engine traffic. Thus, we'll be increasing the influx of users with each contest and self-promoting through them. Unfortunately, I don't know much about how to otherwise promote the contests. Perhaps posts on related (math, physics) site metas? I don't know how appropriate that would be.
I think your suggestions for the first three are great and very appropriate for this site; they are some of the most significant parts of philosophy. For inspiration (though this may be obvious), I would recommend just looking at the most used tags. Some of these are: epistemology, philosophy of science, theology, and political philosophy (I gave examples I think would be good).
Just my thoughts (would you give a penny for them?).