FIFY. (The central government has ministries/部; local governments have departments/局.)
Where did you get that idea? I thought we went over this already.
Buxibans are buxibans, kindergartens are kindergartens, and never the twain shall overlap if they’re doing things by the book.
The prohibition on foreign language lessons is for kindergartens. That one is clear.
The prohibition that buxibans theoretically have is on teaching kindergarten-age kids (teaching them anything), but that one is not completely clear, and the prevailing interpretation seems to be that they can have students in that age group as long as they don’t operate as de facto kindergartens.
Yes, because the standard is, if they’re the right age to be in kindergarten, and they eat and sleep there, it’s a de facto kindergarten, not a buxiban.
That part I have no trouble believing. Also, the kitchen-garage sounds credible to me, not that I would endorse it. 
If you have a kindergarten, you can’t have foreign language classes, period, and you can’t have foreign staff unless they’re exempt from the work permit system and are qualified educare providers by the Taiwanese government’s standards. But if you can find and afford them, then I suppose you can have them interacting with the students and giving them an English language environment, without any actual English classes.
In the EFL industry, deportation is typically caused by a work permit violation. If you’re permit-exempt, there can be no deportation-caused-by-a-work-permit-violation.
If you’re paying bribes, you can be found guilty of a criminal charge (or several), which in some cases may be enough for deportation.
In Taipei, you become a buxiban once your total student population reaches 5 (not necessarily in the same class). The New Taipei DOE buxiban regulations are available somewhere on their website for anyone who cares to dig them up.
Think of what else you could have spent all that time doing. 