recommendations.
Taiwan: has a lot of good food. I would recommend night markets, which have many different varieties of food. I wouldn't recommend going to the top of Type 101, but it is worth seeing. There are also a lot of good hikes in and around Taipei. I'd recommend doing a little research depending on what kind of hikes you like to go on.
South Korea: there is not a ton of things for you to do in half a day. If you really want to experience Korean culture I would recommend going to get a fasule or some sort of skin care place as that is basically Korean culture. I would expressly recommend it if your wife likes skin care. I didn't love Korea and was definitely my least favorite country out of the ones you are going to that felt like it didn't have as much in it.
Japan: Japan has some of the best food and I would recommend trying to find athletic Japanese food. I would recommend finding ramen that is athletic Wich is some of the best food I have had. I would also recommend going to a Kakigori place Which is like a shaved ice but is really big and I would maybe recommend sharing one. If you are going to the temple I would recommend a Tonkotsu that is a little walk from the Temple.