Private schools – not to be confused with buxibans – don’t have an absolute exemption from the LSA, but if they play their cards right they can get away with it. (I’m assuming they haven’t closed the loophole yet.)
As for international schools that qualify as 外國僑民學校 (“schools [or educational institutions, depending on which article of the official translation you use] for international residents”), they need to follow the law of the relevant foreign country.
This misunderstanding is very understandable. 
Laobao has nothing to do with your residential status. Whether an employer needs to register you and pay its part of your premiums (and also deduct your part from your salary) depends on the Labor Insurance Act and relevant official interpretations. Tldr: for most employers, it’s obligatory.
Laobao will not give you much of a pension, but it will give you something (勞工保險老年給付).
Yes.
However…
If you’re a part-time (or unclear status) teacher at one of these schools, check with the labor department.
That iirc is another translation that’s been used for the above definition of “staff”. For ordinary jobs under the LSA (like buxiban teaching), this definition doesn’t apply.
The main discussion of that is here:
Tldr: PR’s and foreign spouses qualify for the new system (LPA) and only qualify for the old system (LSA) if they chose to keep it before the deadline. Other foreigners (normal ARC) qualify for the old system only (last time I checked). This doesn’t apply to all teachers in (non-buxiban) schools.
Some countries might have an escape clause for that, you know, like criminals deducting the cost of guns etc. from their income for tax purposes. 