If everything is arranged beforehand through the school, then you can enter on a student visa. It depends on how much time you have and how the school wants to handle it. Most of my recent classmates arrived on a 90 day visa free and then the school helped them to convert to a student visa.
Note that you are not allowed to work on a visa free entry, and you must wait a period of time on a student visa (6 months? 1 year? Not sure) before you can legally work part time.
Of course, there are students who work part time teaching privately, although illegally. Whether you want to risk fines and/or deportation is up to you. It's not common as far as I know, but it can happen.
I'm not promoting illegal actions here. That said, you'll probably find lots of opportunities to teach illegally, both privately and through buxibans. Read through the experiences here on Formosa of people who have been caught working illegally at kindergartens or without work permits, then decide for yourself if you are willing to risk it.
(A) I'm sure the school will try to fulfill your request for Taiwanese roommates it possible.
(B) I think it depends on your own personality. Have you lived with roommates before? When I considered living in a dorm at Tzu Chi University, I decided that I personally couldn't handle four people to a room, no air conditioning, and no meat on campus.
Short term leases are possible but you have to find them. Being able to speak some Chinese is in your favor. The school may be able to help you, and you can also look on websites like 591.com.tw.
Converting your license depends upon whether or not your country has reciprocity with Taiwan, and even the Department of Motor Vehicles seems to be confused on that point. As an American, I know most state licenses can't be converted. However, an international license can be used in Taiwan for up to one year, I believe, as long as you take it to the Department of Motor Vehicles in Taiwan and register it there.
Scooters are ubiquitous, but I see plenty of larger motorcycles as well. If you have an ARC you can legally register a vehicle in your name. A couple of my friends have had their scooters stolen, but to me it seems like theft is not nearly a problem as it is in the United States, for example.
You're right, whIle a lot of people here are somewhat familiar with Pinyin, they use Zhuyin in day-to-day life. My experience has been that the teaching materials used in language centers generally use both Pinyin and Zhuyin, so no worries there. You can get by without using Zhuyin, but it's pretty easy to learn.
I've only been to Tainan once, just recently, didn't see any gangsters, lol.
My only advice is that if you do come, remember that many Taiwanese judge all foreigners by the actions of a few, so please, please don't be the kind of person that ends up on Apple Daily News. Keep your nose clean and enjoy your time in Taiwan.