Really, don't worry about offending them because it seems that you are very deliberate and thoughtful and there's another standard for American/gaikokujin behavior in Japan anyways. You will be given the same cultural leeway/forgiveness that you gave the host student for certain.
1) I agree with the above posters. Tell the 16 year old that you are planning to pay for the hotel and you would feel uncomfortable if they paid for the digs. If you end up still doing the Asian "who pays for the bill" struggle, and you lose out, don't feel bad. They will feel good that they provided for you and paid you back in some form for what you did for their son. Also, you can bring extra gifts in case you are worried that they will pay and you can feel more relieved.
2) They will ask you what you want to do (guaranteed) but I initially suggest that you ask them what is best. Then, after their reply, you can suggest what you want to do. You are their guests so they will be ok if you have other plans or want to be by yourselves for a part of the stay. Also, they might even have plans of their own as well, so it is good to ask them as well before answering.
3) Regarding Gifts. It's hard now because of globalization and how much you're willing to spend, but I'll take a crack:
- In the past, I would recommend Coach items for women/girls because Coach is much more expensive in Japan, very popular and even in line with Gucci!! They even get more styles in Japan than in America I think but it is much cheaper in America. With Coach's rising prices, not as big as a gap as even two years ago tho..
- For middle-aged women: Le Sport Sac stuff was very popular recently in Japan as well and it is cheaper in America as well. But not sure how popular it is in Japan recently.
- For men: Liquor
- There was a thread on gifts to Japan on flyertalk that talked about sports memorabilia but not sure if it's so good since sports shops are so prevalent in Japan. Seattle Mariners and Ichiro stuff would be obvious but I think that they have it in Japan already most likely..
- Some other things that are good presents for young girls/guys (teenagers/20 year olds): vintage t-shirts that you can get at Salvation Army with American lettering on it or anything with an American college or school on it. These things are expensive in Japan!
- For anyone: maybe University of Washington gear/t-shirt or, Stanford/Harvard clothing.
- Other no-brainers: makeup for women/girls, necktie for the dad, scarf/muffler, etc.
It got pretty long.. my bad.