I'm sure you already know this, but Traditional Chinese is used here in TW. I know it's all relatively the same, but might want to think about what the future boss might say if the task requires you to read/write in traditional.
You are correct, either sometime around Chinese New Year or right after New Years is a good time to look. It depends on when the company pays bonuses or when they have their annual dinner.
Your wife could get a job in a Japanese speaking company since there's plenty of them in the Zhongshan MRT area. She may not have any problems with the job, but might be tough for her to interact with her coworkers.
It will be tough for the both of you if you do not bring anything extra to the table. However, you are fluent in other languages, so that can help! The premise of a company giving you an ARC is if the local is unable to do the job you're applying for.
If you're being transferred here from your current company on a 1-2 year contract, this salary is expected. If you are job hunting, it's unlikely.
As mentioned before, best case scenario is having your current company transfer you. If that is possible, all money pertaining to moving and rent should be covered. However, if you plan on coming on your own, I would save up to 50,000-60,000 NTD for a 1 to 1.5 week trip. This (very rough) estimate includes 1500-2000NT/night hotel + food + transportation + deposit money.
You'll just have to be willing to pay up. A decent 1 bedroom in a new building with security and elevators can run you up to 20-30K no matter where you are. 20-30K may not seem like a lot, but when you compare it to the norm (around 10K) for locals, it's a lot.
If you cannot stand humid weather 365 days a year, you should think twice about your decision to move here. However, you won't have too many issues finding a local brew to cool you down since the craft beer scene is booming.
Food here is nothing like it is at home. For some people, food here is the worst thing in the world. For most people, it's food heaven.
If you and the wifey are looking for that next adventure and Taiwan is the destination, don't let comments on the forums discourage you. If you aren't too too worried about income and just want to live somewhere new, 50-60K a month is pretty comfortable as long as you're watching your spending. If you and the wifey have an income, even better.
I think what you have to worry most about are finding that good paying job and the visa/legal issues that follow, in which @yyy was kind enough to bring up. All the rest, is the thrill of living in a new country, right?
Unless you're teaching at a school or university, teaching at cram schools here is a black hole. I do not need to go into too much detail since there's plenty of threads you can look up here in the forums. You are working professionals with plausible credentials, I'm sure if you try hard enough, you don't have to be willing to teach.
Sorry for the novel, I hope it was helpful!