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? I am currently working at Europe and applying for tenure track positions in US. I will attend an interview that is held by a school. 
The committee chair told me to research on the air tickets. 

Also, in her message she asked if I applied many schools in US and if I plan to fly back and forth. 

I do not know how to answer this question because I am not sure what she meant. Does she want other schools to share the international fare? If so should I contact other schools about it (that was the first school offering me interview)?
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?