Currently I see no arrangement for viewing the list of posts that were closed by a user in his profile. I think there should be a list of these questions in the profile page of a user and details of these posts would only be visible to the user only. Other than that I also suggest a badge system for closing posts. If there is a badge for it, I hope more people would be interested in closing unnecessary posts and searching for them from the archive. Currently only badges for review works are available where we could decide for close votes. But the questions in this section are shown by the system. But I suggest separate badge for closing posts so that people who have close vote privilege feel interested to search and cast close vote on the posts themselves instead of waiting for the posts to come in the review queue. As the site is now growing, I assume the FAQ section may also change. In that case, it would become necessary to search and close previously posted questions as per the new policy.
I don't see any benefit this could have for anyone other than encouraging people to try to close any post they don't like.
Jeff Atwood ( one of the creators of Stack Exchange) said it best (and I don't have a reference at the moment) but it is better to positively encourage people rather than to negatively encourage them.
Closing posts should only be reserved for those that aren't salvageable, are off-topic, and spam (which those get deleted regardless). By giving users a stat and badges for closing, we would be encouraging them to pick out little things that can be fixed as an edit and close the question instead of working with the asker to improve their question or clarify themselves.
There are already badges for flagging posts for mod attention as well as editing badges, I think these serve the purpose well enough for encouragement, and the current moderation team as well as several high rep, active users work well enough to get new users accustomed to the FAQ/ruleset of the SE engine.