The flip side of that is, to get the administrative penalty enforced against the buxiban, if the local labor department doesn’t happen to inspect the buxiban proactively, someone needs to make a complaint.
And the flip-flip side is, of course, the kindergarten situation is illegal for both parties, so how far either party really wants to go is an interesting question. 
Yep. This alien says your employer violated the ESA. 
And even if your only work at the kindergarten had been the special events, it would still probably be deemed work and thus in violation of the ESA.
Your mistake is understandable, but strictly speaking you’re not in the clear. That’s not something your employer can hold against you in a civil suit, but the police can hold it against you, if they want to, and if you file a complaint against your employer, who may be suffering from frog-in-well sydnrome and worried about losing face, it’s not inconceivable that your employer would “cut off its nose to spite its face”, as they say. 
On the other questions:
Tl/dr: buxibans are special business entities that the BLI exempts from the standard 5 or more rule, but if a buxiban already has (or has had) at least one employee with labor insurance in the buxiban’s name, it’s mandatory for all employees, including part-time employees (and foreigners). I guess you can call that the 1 or more rule. 
Longer explanation:
The 5 or more rule in the Labor Insurance Act is for both companies and firms (行號, see LIA Art. 6 Par. 1 Subpar. 2), but the BLI interprets the former as excluding buxibans established by companies and also interprets the latter as meaning only businesses (商業) established in accordance with the Business Registration Act, even though buxibans are still business entities (事業單位) under the Labor Standards Act and what not. The LY’s notes on the LIA don’t really clarify this.
The 5 or more rule does not conform with the ICESCR according to the EY committee that studied the implementation of the “two Covenants” about 10 years ago, but the MOL considers the law impossible to amend until a “consensus” is reached between government, labor, and industry. 
To make things extra confusing, the 1 or more rule is not written into the law itself but is instead found in an official interpretation by the BLI; this interpretation has been endorsed by civil court judgements, and to my knowledge there’s nothing to negate it.
Anyone claiming foreigners aren’t subject to laobao just like Taiwanese are is talking utter 
. See LIA Art. 6 Par. 3.
Correct: your employer is still bound by the LSA, even if there is no laobao, so the “no sick leave pay” clause is void. Also, if your employer is obligated to register you for laobao (see above) but doesn’t, and this results in you failing to receive a benefit to which you should be entitled under the LIA, you are entitled to compensation of exactly the missing amount of the benefit under Civil Code Art. 184. Unfortunately, the labor department and BLI have no enforcement power for this, so you would need to file a civil suit to have the obligation enforced, and in this case it would probably be a very small amount (just the part of sick leave that isn’t payable under the LSA). Even so, if your employer was obligated to register you but didn’t, or did register you but didn’t pay the full premiums owing (didn’t pay at all or didn’t correctly report your salary), the BLI can fine the employer four times the missing amount of the premiums (and it would be ten times the missing amount if you were injured on the job).
Yep, at least some violations there.
As I said, the new rules for holidays are confusing, but the way I remember it, Labor Day (May 1st) is the one extra holiday that “workers” are still entitled to even though public servants are not, so unless a proper rescheduling arrangement was made, you should be entitled to double pay for that day.
A holiday lasts 24 hours (with special rules for night shift workers), so before or after 6pm is neither here nor there.
As always, nothing you read here should be taken as a replacement for proper legal advice.