Taipei

Taipei is "cold" (ha ha ha - I'll explain this in a minute) in December and January. Since Taiwan is a tropical island, "cold" means that the highs might be only 18 or 19 degrees and the lows will be around 13 degrees. Those months are not very rainy by Taiwan standards, so it could be a very good time to go. I'm not sure about this, but I have the impression that it might be windy at this time of year, so you might need a jacket of some kind. You will hear Taiwanese people say in all seriousness that Taipei is cold in the winter, but I've just told you what that really means.

As EU citizens you should be able to stay visa free for up to 30 days. There is no possibility to extend while in Taiwan. I stayed at a pretty expensive hotel in Taipei as a treat to myself, so I'm not a good person for giving you the living costs. Meals are not particularly expensive, depending on where you eat. You could get by for 5-10 Euros for sure at many places or pay more at more expensive restaurants.

I'm a little tired of answering the constant "What can I do in Taipei?" questions we keep getting here. See here for suggestions:
http://wikitravel.org/en/Taipei
If you want to visit a real Taiwanese temple, I highly recommend a visit to the fascinating Longshan temple. Whatever you do, I'm sure you'll have a great time. Taiwanese people are very friendly and it's not difficult to find English speakers there. I haven't been to the other countries you asked about - sorry.

It will be cool/chilly (not that cold,maybe you'll need a sweater but probably not a coat) - it could get as high as 25C or dip as low as 5C in Taipei.

Depending on the country you are from you may not need a visa - you can get one on entry. But each country is different - you can google that answer. Most Europeans shouldn't.

In Taipei you will need about $60USD/day/person on a tight budget. That means youth hostel, public transit and good but not fancy meals. A normal hotel will cost that in room fees alone but a dorm bed in a hostel will save you a lot. If you had a free place to stay (like couchsurfing) you could get by on $30USD/day per person and if you really scrimped (saw temples instead of museums, didn't shop, ate cheap noodles and other small-restaurant fare) you could probably do it on $15USD/day/person (NOT including hotel fees)...but that wouldn't be much fun and is not easy to pull off if you don't know Taipei well.

Lots of stuff to see. I can't even begin to say. Tons. Not trying to boost my own writing but my blog (linked below) covers a lot of tourist-friendly stuff in Taipei.

It's worth keeping in mind that the weather can be very drizzly around this time of year, and sunshine can be rare. Taipei's temperature may sound moderate, but dampness can give the coolness an edge, though it will by no means be cold. If we're lucky this year will like last year have unusually sunny, warm and dry weather.
Agree with Channa about prices. US$30/ day is doable, but no lower.
Taipei and the surrounding area's main features are hiking (featuring views of the city and/ or waterfalls), temples, coastal scenery, world's tallest (completed) building, old Japanese-era towns and mining areas, museums featuring some of the world's best Chinese art. Though Taipei is not a major destination like Beijing, Hong Kong, etc, it has a personality all its own and lots of things worth seeing.

Just stuff a sweatshirt and a light rain jacket in your day pack and you will be fine.

+1 to zhaoyun -- the temperature might not LOOK low, but very high humidity sends it right to the bone. I would wear a thick jacket on colder days in the winter.

was at taipei in january.... piece of cake the weather.

however on yangmingshan, it was 9 degrees celcius and i got cold burns on the face thanks to the wind

checkout my blog:

[http://quirkymalaysia.blogspot.com/2008/02/taipei.html]