O.k., here goes:
Food: no problem, unless you are vegan. Most restaurants offer vegetarian dishes. But beware, chinese vegetarians (buddhists) don’t even eat garlic or chilli … so if you do, let them know.
People: basically two groups: those working in the science park and English teachers. Judging from what you wrote in your post, I’d see you more with the latter group … 
There are a lot of cultural activities in Xinzhu, but the city doesn’t really strike one as being the hippest place if you know what I mean. For some reason the fairly large student community in Xinzhu doesn’t seem to leave its mark as much as you maybe would expect …
Pubs: everywhere downtown, I’d suggest you do a pub-crawl to see everything and take your pick, hard to say really …
Running: Big Problem! To answer your question in one word: No. The air is quite bad more or less everywhere, but there are a few places where you don’t notice this as much: the university campuses of Qinghua & Jiaotung Universities along Guangfu Rd. are adjacent to each other and offer a few o.k. routes that can be quite demanding in profile (i.e. hilly), esp. Qinghua around the lake in the back of the campus. It’s also fairly green there, as Taiwan-standards go …
Another good place is 18 Peak Park (Shiba Jian Shan Gongyuan) also more or less off Guangfu Rd., quite a large Park that’s very shady and has one major route going up and down again the main mountain (and its 18 peaks …) plus quite a few smaller foot-paths and trails that are also good for a cross-country style jogg and are fun to explore. Other than that there’s the Green Grass Lake (Qingcao Hu) close to Minghu Lu that you can run around, but that’s not too exciting. Across the 18 Peak Park, on the other side of Baoshan Rd. there’s the fairly new Hsinchu Botanical Garden, not too big but also a good run with some steep hills & some stairs …
The wheather can get quite warm in the summer, tops out at about 38C and is very humid, the winters stay humid and become what feels a very cold 10C, doesn’t sound like much, but when you’ve got the wind blowing on your house with no heating, no isolation, single-pane windows and tiled floors you’ll probably believe it …
Books: there are a couple of Eslite bookstores downtown, one in the Sogo dept. store and one close to the central round-about (the “Dong Men” (East Gate) Circle, there are concerts on the central “island” of the round-about on weekends in the summer, very mixed though, quality-wise …), plus there is apparently a Caves bookstore now somewhere away from the city center, but you’d have to ask around for it. There are obviously many many more, but these are the ones that carry English and other non-Chinese language books and magazines. Caves is also known for good Chinese language learning material.
Fashion: none that I would be aware of 
Cost of living: now that depends … you can spend up to 100k/month on a house, or you can rent a room in a shared flat for 5k/month; you can eat out every day and buy imported groceries, or you eat a bowl of noodles every day for 50NT$; you can buy an expensive imported car or ride a scooter … it’s all up to you …
Hope this helps, Xpet.