Basically my contract is almost up, but I plan to switch jobs rather than renew.
I already know I need to extend my current ARC to the job search extension. Should I hand in my notice, how long will I have to apply for the extension? Would it be wise to try and apply before I tell my boss I’m leaving? Or do I need the cancel contract agreement to extend?
Ideally I want to hand in my notice ASAP, but I also don’t want to give my current boss the opportunity to screw up my ARC status.
You need to give a certain amount of notice to your employer depending on how long you worked there. 10 to 30 days depending.
You need to sign and get a copy of a Cancel Contract agreement to apply for extension.
Fify. Since OP is on an ARC (not APRC), the contract is not permanent, even if both parties mistakenly think it is. The “10 to 30 days” rule is for permanent (non-fixed term) contracts only.
Would this really hold up if company tried to sue?
Terminating a contract requires a valid reason. If that reason is it’s permitted by the contract or we reached an agreement to terminate, so be it. Otherwise, you need to find another reason.
Is it in the employer’s interest to refuse to let go of someone who doesn’t want to be there? Probably not. But people don’t always do what’s in their interest. 
Call the labor department. You can also look at the English version of Taiwan labor law if you google it. It is unlawful to withhold someone’s Cancel Contract Agreement. The school can be fined.