Hakone/MtFuji/Tokyo Ryokans

Hi all,
After looking through the various recommendations I am still as confused as when I started!

These are my queries, any comments will be appreciated:
1) Take the NeX to shinjuku or JR to Odawara straight?
2) Is it possible to get a ryokan for under 25000 Yen for two with private baths (not attached but available for booking would be sufficient); meals not neccessary.
3) How long does it take to travel from Hakone to Mt Fuji area?
4) Would it make more sense to just stay in Hakone instead of moving to another hotel in Mt Fuji area?

Take the NeX to shinjuku or JR to Odawara straight?

It is your choice:
http://www.japan-guide.com/e/e5206.html

Is it possible to get a ryokan for under 25000 Yen for two with private baths (not attached but available for booking would be sufficient); meals not neccessary.

Yes, there are ryokan at this price range including meals and use of private baths (although not on your room). But I have no specific suggestion.

How long does it take to travel from Hakone to Mt Fuji area?

Hakone can be considered to be part of the Mt. Fuji area. If you mean the Fuji Five Lake area, it takes about two hours:
http://www.japan-guide.com/e/e6905.html?aFROM=5200_6900

Would it make more sense to just stay in Hakone instead of moving to another hotel in Mt Fuji area?

Staying in Hakone is more convenient with respect to luggage. Staying in each area once is more efficient in terms of travel distance and time.

Could someone tell me if it were possible to purchase the JR and Odakyu tickets in singapore?

And would it be feasible to just take a day trip to Nikko from Tokyo?

Thanks for the help!