Is it possible to sit the written exams through CATS then do flight training in Austria? Or is it a better option to do all the training through one country?
If you want to chop and change between EASA/JAR States, it has to be with the agreement of both Authorities, Ref JAR-FCL 1.065. Some are more than ready to "agree", others are not.
In the US you just need a commercial certificate with an instrument rating and enough hours in the logbook to be competitive for the position. Is it similar in the JAA countries?
If you want to exercise privileges as an IRI, you will need over 800 hours under IFR (not to mention the IR on your JAR licence) - ref JAR-FCL 1.395. Instructor jobs for PPL are easier to come by. I can't say exactly how the situation in Austria is, but here you would be joining a bottom heavy job market.
Good luck with whatever you decide and good luck for the move!
RIX