One way from Hakone to Shinjuku

We will travel from Kyoto in the morning to Hakone and stay 2 nights in Hakone. Then we will travel to Shinjuku. Is 3-day Hakone free pass best suitable for us? If so, where should we buy it? what is the difference between purchasing from Shinjuku station and purchasing inside the free area? Thank you.

Yes, it is your best option. You should purchase it at Odawara (or Mishima) Station and pay for the trip back into Tokyo separately. The cost of the pass is outlined here:
http://www.japan-guide.com/e/e2358_008.html

It means purchasing inside the free area does not include a trip from Hakone toTokyo (covers only traveling in Hakone), right? If so, should I buy Hakone free pass at Shinjuku station? (as it includes round-trip from Tokyo to Hakone with additional fee 1,140 yen) so that I don't have to buy a separate ticket from Hakone to Tokyo.

It means purchasing inside the free area does not include a trip from Hakone toTokyo (covers only traveling in Hakone), right?

Right!

If so, should I buy Hakone free pass at Shinjuku station? (as it includes round-trip from Tokyo to Hakone with additional fee 1,140 yen) so that I don't have to buy a separate ticket from Hakone to Tokyo.

No, because you come from Kyoto, it would be a waste of time and money to travel via Shinjuku.

Thank you Uji!!

My itinerary is Tokyo-Kyoto-Hakone-Tokyo. Before going to Kyoto, I will stay a few days in Tokyo so I can easily buy a ticket at Shinjuku station. By the way, what kind of transportation is the best for one-way trip from Hakone to Tokyo?

Can the Hakone free pass inclusive of round trip to Tokyo be purchased in Hakone? Is there only Shinjuku station were I can buy such pass?