Yes, they do. Your wife will have to ask them, and they will charge NT$1000~NT$2000 depending on size and type of aircon unit.
The big department stores usually have appliances not available in places like carrefour, I am thinking of dishwashers, big ovens and the like. However, they are more expensive, and most of the appliances they sell are the same as the ones you can get in carrefour, Geant RT mart and the like. The warranty should be the same. I have had warranties for cheap DVD players and the like honored in the local Geant.
You will find more different cars available in Taipei as more people live there.
However, as you mention the words “small” & “inexpensive”, then you should be thinking in the lines of 4 door saloon of local make. It has to be made here, as spares then become cheaper. A car like that should be possible to get in Hualian. I drive Nissan, the Toyotas are a bit more expensive but are said to last longer, the Hondas are OK too.
Try to go [url=The hidden costs of owning a car for more and better answers. Michael J Botti (a neighbor) knows a lot about this, and reading his and bens posts would be a good start.