Problems with the current format in no particular order:
-You don't seem to have made up your mind whether this game is meant to be for individuals or a team effort. The existence of a prize defines this as an individual contest. In rule suggestion B you say "This is to encourage people to ask questions that help everyone, rather than selfishly trying to win with a lucky guess", but why shouldn't we be selfish? Prize only goes to one person. Logically, I want to take whichever course of action benefits me the most and everyone else the least. However, that isn't possible because all info is public. If you wanted it to be a team exercise (presumably trying to guess the skill with the least number of questions), your format doesn't work either - our success will be defined by which questions you choose to answer. In either case it doesn't matter if someone asks a really clever question - from an individual perspective it doesn't help because everyone learns the answer and from a team perspective it may not help because you may pick a less efficient question. You need to clearly decide whether this is an individual or team and define the rules for that purpose.
-Due to the number of skills in the game we have no choice but to ask the same narrowing down questions every time. The chance of guessing the skill correctly without having had 8-12 questions answered is so remote it isn't worth trying. It's going to take a week and a half before it actually gets interesting. Furthermore, given you only answer one question a day, we might as well just leave the narrowing down to one person and then all jump on board once the possible options are down to a dozen or so. If you want to keep a similar format to what you have currently, what you should do is to choose information to reveal each day and leave player interaction solely as guesses. This would also give you the freedom to provide as much or as little information as you want. You could also do something a little more abstract, for example make statements such as "these two guesses have something (the same thing) in common with the answer". That would make it more of a puzzle as it would be up to each person to determine what the clue is referring to. You're still going to have the above problem though.
-The possibility that someone could correctly guess the answer first and then not win is utterly nonsensical. You must confirm guesses either way at the first opportunity, otherwise it just becomes a matter of favouritism. The person who wins is decided entirely by you, as opposed to solely the merits of the participants. However, this biases the competition to people with a similar availability time to you. You could wait until 24h from your previous post and then pick randomly (via a random number generator) from the correct answers if multiple people get it right.
Food for thought:
Games of this format fall onto what's known as a skill-luck matrix. Chess, for example, is a high skill-no luck game and poker is a high skill-high luck game. Both of these are played competitively at a high level. Your format is a no skill-moderate luck game, which by definition are uninteresting. This needs to be remedied one way or another. Here's how you could create a high skill, moderate luck game: keep everything strictly individual. Users ask questions/guess solely via PM and you also respond via PM. The objective is to guess the skill with the fewest questions. This alleviates the above issues entirely. Downside is that everything happens behind closed doors so while it's a much better game it doesn't really work as well as a forum game.