[MMM]

what are some awesome places I should visit during my stay?

I don't really know Tokyo that well. Hopefully someone here can help. What sorts of things are you interested in?

The place I am staying it regularly reaches around 80-90% humidity; is the same true for Japan during summer?

Yes.

How are the public drinking water locations in Japan? In Korea, it's very rare for me to find a drinking station in a public setting- is it also true for Japan?

Good question. Not very common, I think. You can see drinking fountains in large public areas like shopping malls, but vending machines are everywhere.

Would I have to carry a handkerchief with me or is the situation in Japan a bit more technologically up to date so that I wouldn't have to resort to carrying handkerchiefs to dry my hands?

Most people carry a handkerchief. A lot of public restrooms have paper towels, but many have air dryers (that tend to be very powerful).