I suggest that you stay near the Guting/Shi-Da neighborhood or near the Red/Green subway line. That way you can go to the big tourist night market (Shilin) easily and you can also check out the Shi-Da night market which is a bit different. And you can go to Yongkang St. and check cafes etc. there. Plus ice monster.
For 民歌, your best bet is probably 女巫店 (Witchhouse). They have live music Thurs-Sat. starting at 9:30. http://www.witchhouse.org/
. Also check 河岸留言 but they tend to have more jazz than Minge.
It will be difficult to find a place with nice view and fresh air in Taipei City. For that, you need to leave Taipei. Hualian/Taidong are the best.