Bigger city [population wise] = more schools = more jobs. But does not = higher pay. Pay is pretty much the same all over the island. The only difference is cost of living and Taipei is high.
The rule of thumb is the bigger the shithole the more jobs are available. Taoyuan and Hsinchu are top contenders. IMO Taoyuan is the undisputed champion on account of lower living costs. If you’re looking at the long term then I would recommend Taipei. Most of the top gigs are there, but not many can have them or even know about them.
I’m going to gently disagree with you on this. OK, pay is generally the same in terms of 1000NTD an hour isn’t really much different to 600NTD an hour. Nobody is retiring early on either of those. However, if somebody is earning 600NTD an hour and wants to up it to closer to 1000NTD an hour then northern Taiwan is the place to go.
Salaried jobs (uni, public schools etc) are there or there abouts the same across the island. Funnily enough they could well be higher in places like Penghu.
EDIT: I’m assuming the OP is looking for teaching jobs. Apologies if not.