Taiwan gives 30 day landing visa to citizens of a great many countries (UK citizens get 90 day landing visa). If you get an Alien Resident Certificate by virtue of working for Ci or EVA then your WIFE (but not a girlfriend) could stay in Taiwan by piggybacking on your ARC. You would both be entitled to national health care after a short qualifying period. But she wouldnt be able to work on her ARC thats tied to yours. If she holds a passport from an English speaking country (such as USA/Canada/UK/Australia/New Zealand/South Africa) then she can teach English at cram schools , provided she has a 4 year college degree or a 2 year degree and a teaching certificate . She will then get her own work related ARC based on her teaching English. This is what your girlfriend can also do if she qualifies. Pay is around 20 usd/hour.
Not a whole lot of other jobs. IF she wants to work in another field she will need at least two years experience in that field and be sponsored by a company that is qualified to hire her.
I gather in the past many young pilots have gone to Ci and EVA to get the rating and hours and then left for better positions elsewhere. To stay for decades in Taiwan would mean you REALLY want to be there.
Taipei American School (your child needs to have a US passport to enroll there) and Taipei European School (your child needs a European passport to enroll there) are both very expensive at bout 12,000 us/annum or thereabouts. So if you have children, you may want to let them go to local schools and become fluent in Mandarin.
Please correct me if I am wrong on any of the above, been ten years since I left the island.
edit: Wanted to add that another way for your Girlfriend to stay in Taiwan longer then 30 days each time would be if she enrolls in a Mandarin class. She can get an ARC based on that. There are requirements for how many hours a week she needs to go to school. But its not very expensive and there are many qualified schools.
p.s. A great forum to get info about life in Taiwan is
Taiwan's Global Online Community • Forumosa You can ask the expats on that forum questions and get some straight answers.