Odawara is boring. I hear it is a nice facility, but you'll be missing a lot of must-do's if you spend your entire week in Odawara. It is essentially a sleepy country town. My grandparents lived there.

Must-do's near Osaka include Himeji Castle, and at least a few days in Kyoto (kinkaku-ji, kiyomizu-dera, and gion). Osaka itself isn't that interesting, though.

Must-do's in Tokyo are really hard because everyone has a different idea of what Tokyo is. My ideal itinerary would include odaiba, harajuku/aoyama, and shinjuku. Some like asakusa, and others like roppongi.

While traveling between Osaka and Tokyo, you will want to spend a night in Hakone. You will not need a rental car there, as trains and buses cover all the sites. There are many ryokans with private roten-buro hotsprings, where you can soak without getting intimidated by the locals.