Welcome to the forum luna!
Yeah, plenty of stuff is open - smaller mom & pop places, slightly less likely, but for chains there’s not much of a difference from regular opening hours. See this thread for more information:
That’s a tougher question. I suspect a lot of galleries and museums do close, especially for the first few days, even if the department stores remain open. As posted up-thread, I personally think trying to leave Taipei and deal with the ensuing traffic is an awful idea. For actual CNY festivities, I don’t recall ever seeing anything (it’d be a bit like looking for fun on Christmas Day in a North American city - you’re not going to find much); Lantern Festival, officially two weeks later but with lantern displays often starting earlier, is more entertaining. It looks like the biggest event is in Taichung this year (I think it moves around from year to year?), but there should be smaller displays at a few locations around Taipei.
You could always visit Dihua Street for the pre-CNY shopping. If you’re new to Taiwan, I can see that being interesting.
Traditionally the weather is dire at this time of year, but if the current fantastic weather holds, you can go hiking or enjoy the riverside cycling trails or whatever other outdoor activities you’re interested in.