A question in the future research section - is it considered appropriate to write a paper on it? I recently read a paper, which posed some interesting questions in a section named further research. 

They concluded the section by writing, "we leave all these questions for our future research". 

I am really interested in one of the questions, is it considered inappropriate to write a paper handling one of those questions? The pronoun "our" indicates that the author wish to dig in to this question himself. However the paper was published in 2012, and by browsing the authors personal webpage I get no sign of a paper like that in progress.

In case I wish to write a paper on that particular topic, should I notify the author who posed the question? (Say, by asking, is question X still open?)
Just because an author puts forth a question doesn't mean that she or he owns that area of science.  You are free to pursue any question that was posed in or inspired by an article.  Even if the authors are actively pursuing the question, your research on the question will likely take a different form.  It is unlikely that you will have exactly the same methodology, so the results will be different.  And, even if the studies are exactly the same, the contribution you are making is important -- i.e., replication is essential to the advancement of science.  

So, if you are inspired to pursue a given research question, full speed ahead!