1. By Keisei Line train. Skyliner for 1,920 yen, limited express for 1,000 yen. Both take 1 hour, Skyliner a little faster.
http://www.keisei.co.jp/keisei/tetudou/keisei_us/top.html2.Odakyu limited express "Romance Car" would be the best. The journey to Hakone-Yumoto is around 1 hour and a half.
http://www.odakyu.jp/english/rc/index.htmlCosts It costs 1,720 yen to Odawara, then you might want to make good use of the two-day "Hakone Free Pass", which costs 3,900 yen (from Odawara). Should be bought at Shinjuku before the journey.
http://www.hakonenavi.jp/english/freepass/index.html3. You might catch one of the Hikari trains at Odawara, which leave 7:10hrs, 9:10, 11:10, 14:10, 16:10 and 18:10, and takes 2hours 26 minutes to Shin-Osaka. To get to Osaka Station, catch a local train and alight at the next stop. Costs about 12,270 yen for a seat with assignment.
4. Take a JR Special Rapid train, or Hankyu limited express train. The former takes 20minutes, costs 390 yen, runs every 15 minutes, the latter takes 27 minutes, 310 yen, runs every 10 minutes. Then transfer to the Port Liner. Alight at Port Terminal. Five minutes, 200 yen.
5. Refer to the site.
http://www.japanrailpass.net/eng/en003.htmlPlease note that if you're flying into Narita and out of Osaka Kansai, purchasing a JR pass doesn't pay. If you are flying into and out of Narita, buying the pass would be a good idea. So, it depends, and some of my advice needs to be revised in case you buy the pass.
6. In your case, not recommended. There are cheapish tour bus services between Tokyo and Osaka from 3,900 yen, but that means you need to backtrack to Tokyo from Hakone. If you bother to do so, it will work out cheaper than the Shinkansen.
http://www.123bus.net/Happy travels.