I maybe late with answering your query but I hope it will give light to others who have similar issues. We hoped we knew this process when we got our Taiwan ROC passports 2 years ago but oh well better late than never.
I am answering your questions backwards. 2. does she need taiwan visa?
Yes. If you are a Taiwan ROC passport holder what you need to apply for is the entry/exit permit visa from TECO Philippines, which is at Window 10. Get a number from the guard outside the room! Requirements would be an NSO birth certificate and a filled-up entry/exit form with pictures. There is no charge for this, gratis yey! Validity of permit is 6 months and maximum stay would be 3 months. They will take your Taiwan ROC passport since it is where they are going to stick your permit. Processing time would be 4 working days.1. which passport should she use in leaving ph entering taiwan, and leaving taiwan and entering ph?
Bring both of your passports. Show only the Taiwan ROC passport and then show your Philippine passport when asked. The Philippines Immigration officer stamped our Taiwan ROC passport because that is where our visa is but the officer also asked for our Philippine passports probably as proofs that we legally stayed in the Philippines and not as illegal immigrants(?). When we reached Taiwan airport, we just showed them our Taiwan ROC passports as well and they didn't ask for our Philippine passports. They scanned the bar code on the permit and then stamped it with the date of our arrival.
Hope that helps!