Your kids also need to learn Chinese. They won’t pick up a language so much, if they don’t go to nursery but stay home with their mother.
I think you can search on those info in this forum.
One of latest is this.
you can work at more than 1 Buxiban with a work permit for each work place. Additional work without a work permit is illegal.
What does the school mean? There are no distinguish between kindergarten and preschool here, and youngest they accept are 2 yo.
English is an official language of Philippine. So, if she has a univ. degree accredited by taiwan gov. or more than 18 or 20 months of higher education and TEFL, she can teach English.
The job should be a professional one for that employer can get a work permit of your wife. You can check in this site what kind of jobs are allowed for foreigners.
Foreign Professionals to Work in Taiwan https://ezworktaiwan.wda.gov.tw/ezworken/home.jsp?pageno=201508070001
to teach at Buxiban, I think it is not a so big factor. If you can get a teacher license in 2 years before you move, it may be more useful in Taiwan.
Why don’t you try to find a job of your profession before you move? It might be easier than to look for it as an English teacher in Taiwan.