The Lai Lai Sheraton is a great hotel, but it is the only one out of those three that I've stayed at personally. In general, my experience with the "major chain" hotels in Taipei has been overwhelmingly good. The service is excellent, and as a matter of fact, I feel that they treat you like a king whether or not you have any status in the frequent guest programs. That is something that I can't always say for their American/European counterparts, where the metal on your membership card makes you king or pauper... I don't remember where the locations were because I went everywhere by taxi anyway.

What to do depends on what you're into. There's all kinds of museums in and around Taipei; there are night markets which are open air markets that are my favorite memory of Taiwan... just throngs and throngs of people out for no better reason than to hang out with throngs and throngs of other people. There's all kinds of food, though I imagine most Americans would cringe at the sanitary standards of most of the "roadside" stands or places at the night market... but I've never had a problem. If you're into a hike and a climb, take a day trip down to Ali Shan. Mandarin is not absolutely necessary, although it would help a great deal if you had someone write vital comments down for you. If you stay at any of those hotels, the taxi drivers will know their English names.

I can't help you with airport transport because I either get picked up or someone arranges something for me so I don't have to deal with that. The airport is quite a ways from downtown Taipei, though, so I imagine the prices will reflect this.