Sounds to me like you want something like a philosophical chat room. I don't know of any of these, but I would probably avoid them even if I did know of them. My experience with this is that too many people out there are too often to "discuss philosophy" when they haven't much of a clue as to what it is. But you don't have to share my prejudices!

The philosopher Benson Mates once characterized philosophy as a field whose problems are unsolvable. This has often been taken to mean that there can be no progress in philosophy as there is in mathematics or science. But I believe that solutions are always parts of theories, hence that acceptance of a solution requires commitment to a theory. Progress can be had in philosophy in the same way as in mathematics and science by knowing what commitments are needed for solutions. In a sense, this means that sometimes philosophy "progresses" backwards, by coming to understand what extra assumptions are needed to solve its problems. (I've written about this in a technical paper--"Unsolvable Problems and Philosophical Progress" (American Philosophical Quarterly 1982)--and in an essay for a non-technical audience--"Can Philosophy Solve Its Own Problems?" (SUNY News 1984). There was also a recent symposium on this topic at Harvard, and some of the talks from that symposium can be Googled online.

It might not disqualify you at some programs, but it will certainly count against you at most. The writing sample is the primary way of distinguishing applicants' philosophical talents, at least once they have been narrowed down using other criteria (such as coursework in philosophy and grades, letters--though for the competitive candidates, they tend to be equally gushing--and perhaps GRE). A 5-page sample is unlikely to provide evidence that you can develop an argument responding to a particular position that you have adequately and charitably explained. (Of course, Gettier's famous paper is quite short!)

I say all this with empathy--I was a philosophy minor (not major) and did not have a good, long piece of writing to submit when I applied to grad school. I had to use a mediocre, long piece, and was lucky to be accepted in the few places I was. But that was (too) many years ago when the competition was a little less fierce. I would try to work with one of your professors to develop one of your short papers into something more substantial (12-18 pages).

(On the other hand, people should NOT submit pieces longer than 20 pages.)

Great question! The journal Teaching Philosophy has been publishing for decades on different ways to best teach philosophy, and that journal might be a gold mine for you in terms of creative ideas. One of the latest developments is that a great deal of philosophers have been bringing philosophy into play with popular culture. William Irwin had done a great deal on this with Blackwell (now Wiley-Blackwell publishers) and he has an edited volume that brings together some of the best work on all this. I am not sure whether specific exercises are employed, but the major series he edits, and similar series with Open Court Publishers and the University Press of Kentucky might be excellent resources. OneWord Press (UK) has one, probably two books that offer philosophical puzzles to ponder: approximate title, Why is it wrong to eat people? Ted Honderich has a textbook on thought experiments with great questions for students. And you might even check out Gareth Matthews work on philosophy for children. I realize you are addressing young adults, but Matthews has great puzzles for people of any age. Every possible good wish in this for you and your students!

As hard as it was for your friend to explain his dissatisfaction with philosophy, it's even harder for me to be confident that I really understand just what his complaint is. But I do think that a philosophical work fails to a significant degree if an intelligent reader comes away feeling that the author missed the boat or can't speak to anyone but specialists. Indeed, I support this website because I think philosophy is far too important not to be better understood by the general public. I'd encourage your friend to try harder to articulate the reasons he came away dissatisfied by this or that particular work of philosophy: to translate his feelings as carefully as he can into reasons and examine them. That exercise by itself very much counts as philosophizing.

From time to time (although not as often as the public probably supposes), contemporary academic philosophers take up an age-old human concern like the meaning of life. An outstanding example, if by no means always easy reading, is Professor Thomas Nagel's article "The Absurd," originally in the Journal of Philosophy (1971) and widely anthologized since then. If your friend has read that article, does he think Nagel is just missing the point? If I may, I've linked below to a couple of my own humble attempts to explore such issues in a widely accessible way. I hope they're written so that any readers who find them dissatisfying can pinpoint pretty clearly why they do.

Maitzen, "On God and Our Ultimate Purpose," Free Inquiry 31:2 (2011)
Maitzen, "Does God Destroy Our Duty of Compassion?" Free Inquiry 30:6 (2010)

Sure! I highly recommend the Stanford Encyclopedia of Philosophy. There's a link on the "Related Sites" section of this website.

As for the first question, I do (as it happens) work on a college campus in which my office is in an ivory covered building with a tower, and there are some Greek sculpture here and there on my floor, though the most common things (except for other professors, students, books, furniture) in our department are dozens and dozens of owls (symbol or wisdom), owl statues or as dolls, etc. But speaking to the ivory tower as a metaphor, I think philosophers today and certainly at many points historically, very much engage the world and culture at large. Socrates did philosophy at the market place, and now there are many philosophers who seek to engage others through popular culture, their courses that involve very practical moral concerns (e.g. bio-medical ethics, environmental ethics, courses on just war theory and so on), and in publications that have a wide, educated readership (e.g. New York Review of Books, Times Literary Supplement, and so on).

On the second point, I think it is rare to find a philosopher who is strong in logic but weak in math or vice versa. Both do employ some level of abstraction and formality that make the two areas good, if not overlapping neighbors. As for philosophers of language, some have strong backgrounds in logic and math, but I do not think this is as obvious. Philosophers of language are sometimes impressed by the vagueness of our terms and modes of references; to be sure, they want to be as clear as possible about the nature and scope of vagueness (a popular topic at the moment), but philosophers of language as well as those in logic sometimes make a point of recognizing when clarity (you refer to "clarity of mind or perspective in observing the world") is elusive. There is even an area of logic called "Fuzzy Logic" that addresses what some call "fuzzy sets." In classical logic, there is a tendency to adopt the law of excluded middle (everything is either A or not-A), but in more modern times some of us have come to see that an object might be a member of some set to some degree, and this is not an all or nothing matter. For an interesting book that argues that vagueness is a matter of our ignorance, see T. Williamson's Vagueness (Routledge 1994).

I appreciate your appreciation for Whitehead's observation, which I share. I might only add that the increased mathematization may sometimes be a reflection of more precise ways of mapping out a world that could turn out to be indeterminate, at least at the sub-atomic level, and resistant to certain predictions. In a word (well, actually in several words), we may need more math in order to think probabilistically rather than to think in ways in which we could predict with iron clad certainty the ways of the world. (I am not suggesting you disagree, just adding a thought which I hope might stimulate further thinking.)

Maybe to connect the various topics your questions raise: I suggest that it is because philosophers today tend not to be in (metaphorically) ivory towers, but connected with current science, events, issues, that the task(s) of philosophy are so exciting. We want to understand logic, math, language, science, as it is actually practiced as well to explore the implications of such practices and future developments. Also, philosophy often seeks to be integrative or to explore integration: how might different modes of inquiry (math, logic, natural and social sciences, the humanities) interrelate?

Good wishes in your own inquiries!

I believe that in some bookstores "metaphysics" is used to classify books that are "new age," but, technically, in philosophy or for most philosophers, the domain of metaphysics refers to theories of what exist. In this usage, metaphysics is hard to avoid. For a good defense of this outlook, see E.J. Lowe's The Possibility of Metaphysics. Some metaphysical systems accept what may be called the "supernatural" (God, the soul), other systems of metaphysics may be pretty materialistic, e.g. the view that the only thing that exists are matter and energy. Accounts of what can be demonstrated rationally concerns the domain of epistemology (theories about knowledge and what can be known or justifiably believed). I personally think there are good, rational reasons for accepting some forms of metaphysics and rejecting others. For a good book on this, see Lowe's follow up book Metaphysics or Michael Loux's Metaphysics, or the Routledge Companion to Metaphysics (which just came out in paperback).

Emotion is very important in all human activities, philosophy included. Many arguments are fashioned as they are not in a disinterested desire to attain the truth but to get promoted, impress a potential partner, do down an enemy and so on. These might be regarded as cynical comments but then philosophers are usually human beings and they will then have human emotions.

One does sometimes just not like a theory and then we look for reasons of a logical kind to disprove it. It could be something about it that raises our suspicions, like someone we dislike propounding it, for example. One thing I have noted is that few philosophers like to admit they have changed their minds on an issue, we normally stick to our guns throughout our careers. After all, consistency is supposed to be a logical virtue.

Readers might be interested in some of the attempts by philosophers to explain their work, their problems, their philosophical passions to a non-professional audience that have appeared in The New York Times blog "The Stone": http://opinionator.blogs.nytimes.com/category/the-stone/.