What does it mean if a committee chair asks if I applied to many similar jobs and if I plan to fly back and forth?
Unfortunately there are budget constraints at all institutions, and interviewing faculty can be an expensive affair.  Most likely, this is a polite way of trying to figure out whether you'll already be state side during a date range they'd want to interview you. No reason to have you hop the pond repeatedly if the timing works out.  

No, you should not preempt the question and ask other schools about sharing airfare.

I am tempted to think this school is reluctant to spend the money to bring people from abroad over.  It is somewhat strange they would have you research airfare. In my experience, there is staff that takes care of that. I, for one, would mark this in the 'con' column, because if they skimp on airfare for desirable faculty, what else might they skimp on?