If you can’t find a valid reason for early termination, then that’s the type of penalty that would be enforceable if they were to sue you, though they wouldn’t necessarily bother. A quick search will reveal several past discussions about this subject.
First question, is the typical year long Bu Xi Ban contract (with opportunity for renewal) considered a fixed term contract or a non-fixed term contract? I’ve seen advice suggesting both.
It’s a fixed term contract. Even if it claims to be permanent, as a foreigner you don’t qualify for a permanent contract unless you’re a permanent resident, so the length of the contract is automatically shortened to whatever the length of your approved residency is.
If your contract gets renewed within X amount of time after termination, you are considered to be still employed, i.e. you keep your seniority, which makes a difference if you qualify for severance pay (which you don’t unless you perform a valid Art. 14 termination or the employer performs a valid Art. 11 or 13 termination), but it’s still a fixed term contract in the eyes of the law.
Second, given the type of contract I have, how much notice am I legally required to give? I’ve seen different guidelines based on the Labor Standards Act, but what I saw specified only when the contract was terminated after at least 3 months. What If I terminate the contract before 3 months of work? Can my employer legally refuse to terminate the contract?
You can’t terminate the contract, unless you have a valid reason for an Art. 14 termination (in which case there’s no notice period anyway). The other threads explain it in more detail.
If both parties agree to the termination, that’s fine, but it sounds like you won’t get that agreement without paying the penalty.
Third, I really want to look for another teaching job in Taiwan, will I have any trouble getting the 6 month ARC extension given that I JUST got my ARC and only just started this job? I’m not sure how immigration will look upon this, or if they won’t care as long as I have my employment termination agreement.
As long as the contract is legally terminated, and you apply for the six month extension right away, you should qualify like anyone else.