Where can I exchange TWD to USD before leaving Taiwan? Is doing this at TPE airport a good option (I've heard the exhchange rates there are good, unlike other airports), or is it better to go to a bank in Taipei? In the latter case, does any bank exchange money? I haven't seen any currency exchange points (in contrast to Hong Kong where they are plentiful).
At a bank. There are at least two in the airport. Bank of Taiwan is government owned and gives a fair exchange rate, so if you are going to the airport, no need to stop by a bank elsewhere on the way.