Considering moving the family to Taiwan...Looking for some info

You should contact the British and other international schools in Taiwan.
The available places for teachers is limited and highly competitive.

The local healthcare system is great but does not cover everything. Even then if you pay privately it is still cheap compared to other countries.

Your husband would need to get a work permit himself if he wanted to work as dependent spouses do not have work rights. As he has a degree he would get working teaching but there are other areas of work that many expats hold in Taiwan.

Your choices of Taiwan, Japan and Singapore are very good, although Singapore is the "ultimate" expat location with the best conditions, medical and salary, but finding a job there is very competitive. Even if your husband got his CELTA or other TEFL certificate there really isn't much chance of working for say the British Council, who I believe have recently been letting staff go. He's need years of teaching experience and probably with the BC so I would forget about Singapore.

For Taiwan, I would suggest he get his CELTA certificate which is just a month long course. Taiwan has a lot of English language schools and being a native English speaker helps a lot. He is not going to get a job as a plumber so I'd forget that as an option. Also, while plumbers can earn well in the west, they are not so well paid in Asia and he'd probably have much difficulty in being able to work as one, and I assume he doesn't speak Mandarin. So forget plumbing.

For expat packages, you really need to ask the school about what they are offering since they would have absolutely no difficulties in finding teachers to fill a position. Certainly it should include medical and housing or at least a good enough salary to afford decent housing.

Salaries are always good in International schools and certainly enough to have a decent lifestyle. The reason international schools are so expensive is because of the salaries they pay to their staff and the usually decent housing allowances.

Medical is excellent in Taiwan whether covered or paying yourself. The standard is high.