I never take more than a rollaboard plus a small laptop backpack for up to 3 weeks in Japan. To make that work, I book at least one hotel during the trip that has a coin laundry on site. For 6 days, I'd think you could rinse out a few things in the sink if no laundromat is available. The one kicker here is that you are traveling in the dead of winter and will likely need heavier clothing. Check hotels for laundry facilities and plan on a laundry night and you will be fine.

In addition to prescription meds, I always take small quantities of cold-related OTC medications and have never been questioned. Perhaps I just look trustworthy but in many trips to Japan, I have never even had my luggage opened for inspection. They just wave me through. Of course, I look like someone's grandfather and it might be a bit different for college aged girls but probably not much. However it goes, they will be polite. Expect more unpleasantness from the US folks when you return.

It will likely be colder in Kyoto than Tokyo but neither will be arctic. Whoever has been telling you "unbelievably cold" has been exaggerating.