I am no expert, I just happen to have set up many companies in Taiwan and around the world without using CPAs so I know how it works behind the scenes. My advice is worth exactly what you are paying for it. With that said..
guillaumimoso wrote:2. Set up a branch office of my Singapore-based business in Taiwan.
That would be interesting, since I think there's a double-taxation agreement between Singapore and Taiwan. However, does that allow me to get a work permit?
Would I also be eligible for permanent residency after 5 years of doing that? It all depends on whether I like it enough in Taiwan to stay permanently, but who knows? Better to plan ahead. ;-)
Depending on what you are doing you might not even need a branch office. A branch office is good because you get the benefit of an ARC with no minimum capital or earnings, but you can issue invoices and generally make sales as a normal Taiwanese company. The tax rate is also attractive.
If you are strictly selling outside Taiwan then you can just get an ARC through a rep. office from your Singapore company. You will not be able to legally make any sales here.
Either of these ways will make you eligible for PR or citizenship.
3. Given I want to study Chinese, I could stay on a student visa, and work on the side. Translation is an internet-based business, and I wouldn't be called for interpreting jobs in Taiwan, as I don't have Chinese in my working languages (I would do that overseas). Thing is, technically, I'm working in Taiwan without a work permit, so I guess it's illegal, even though I can keep a low profile, but... I'm not very comfortable with that idea. Plus, I guess I can't get the PR status after being a student?
I considered that option way back when but rejected it because I knew I would eventually want citizenship and time on a student ARC does not count. If that's not important for you and you are selling a service to companies outside Taiwan then it would be extremely unlikely that you encounter any trouble. Mainly you should be worried if you are earning a high income and not paying tax.