I think city shops should be open. Not sure about Ah zong mian xian though. actually, i nvr really pay attention during CNY if their shops are open or not. but u can definitely find food.
Taipei to Taizhong by bus is about 3hrs. by train also thereabouts depending on the type of train u choose. by plane is 30minutes. by car should be about 2hrs.
amount of $$$ to bring is highly dependent on individual. but u can note that cost of living there is similar to singapore.
yes, winter clothings are nicer & cheaper there if you go for no-brands.
as to whether cold or not, if u stick to cities, it would not be *that* cold. only if u suay to meet cold-front, then will feel colder. but otherwise, is actually quite fine if you wear a thicker cloth + jacket. but if u intend to go alishan during this period, temp is in the single digits or even zero. if u go hehuanshan, *may* snow only if u're lucky. else is about zero degree. other parts of taiwan towards the south, not cold.
so starz, one winter coat is v. sufficient.