Hello, welcome to Taiwan and I hope you will have a nice trip here. :)
Suggestions:
1. Well, it depends on how much you want to explore here.
But as the first-time visited, maybe a week is enough. You can spend 3 days in Taipei, two days in both Taichung and Kaoshiung. Take the HSR(something like 新幹線), it's cheaper than in JP and can save you much time.
2. This is a really difficult question , but I think you need about 10000yen per day (includes meals, hotels, transportation , ect. But doesn't include airplane ticket).
But I am sure that you can spend less if you want.
miner8.com/ja/83206 this is a website talking about the commodity price in Taiwan.
3. Well, I think you don't need to worry a lot. Just remember to rent a sum card with internet and everything would be fine I believe.
People here can speak basic English and some of us can speak Japanese, too.
So don't worry! Just come visit here with an exciting mood!