Can't answer a lot of these questions unfortunately... But as far as Health-Sports Day (Taiiku no hi) goes, I don't know that you're really going to come up with anything in terms of any exciting event you can participate in. To me, it's just a day off and the significance to a visitor would be to plan things accordingly (ie, certain attractions may be a lot more crowded than a typical Monday).

You'll find "gotouchi" kit-kats (kit-kat flavoured after something that's unique to that region) all around Jpn. But each place will only stock the gotouchi kit-kat specific to that area. However, NRT and TYO carry wide variety, so the airport may be the best places to buy as many flavours as possible in one setting.