May I ask questions that I know the answer to, but are puzzles? For example, can I ask whether the level of water in a cup of ice water goes up or down while the ice melts? This is a standard conceptual question in introductory physics and I know the answer, but can I ask such things (especially novel examples of this sort of question) just to promote people thinking about it and coming up with solutions?
Note that we had Mechanics around a rail tank wagon which is more or less a puzzle question, but was sufficiently challenging to keep several quite good physicist arguing and muddling their way through before we arrived at an answer that I believe to be correct.
I think puzzles would be great for the site.
"Code golf" questions pop on stackoverflow all the time. These questions challenge users to write code that successfully accomplishes a task in as few characters as possible. For example, Write the shortest program that implements John H. Conway's Game of Life cellular automaton. (Mathematica can do this in 151 characters, amazingly!)
This sort of thing actually can be very educational and interesting. Even seasoned programmers can learn a lot from seeing other people try to work out solutions to problems in novel ways.
Similarly, physicists shouldn't get too haughty and start to believe that they know everything, and certainly not that they know every method. You can learn a lot from seeing how other people approach problems. Mechanics around a rail tank wagon is an excellent example of this.
I'd say no, don't ask a question to which you already know the answer. Not because there's some fundamental problem with doing so, but because if you know the answer to your own question it typically winds up being phrased badly. Also, if you ask something like your example, whether the level of water goes up or down as ice melts, you're going to get answers targeted at someone who legitimately doesn't know what the answer is, and they'll sound condescending.
If you think you know the answer to a problem but you're not entirely convinced, that's a different story, because in that case the real question you're asking is "can anyone provide a more convincing justification for this?" And that's a question you don't know the answer to. Those sorts of things can be decent puzzle questions.
I think we already have an abundance of people who have clever things to say, whether or not those observations are related in any way to the discussion they are joining. This sort of thing will get out of control if they are encouraged to show their brilliance in the form of a puzzle. For an example where this can lead you can look at this gem.