1 day tour in Hakone

Thanks Peter for the prompt reply. Based on your recommendation, I had drafted a plan. Can you please kindly advise if the time schedule is achievable as we do not wish to stay for too long due to the cool weather in thhe evening.

1)Romancecar-Shinjuku to Hakone Yumoto (9:27am to 10:58am) ・1 1/2hr
2) Hakone-Tozan Mountain Railway - HakoneYumoto to Gora. (40 minutes)
3) Hakone Tozan Cablecar - Gora to Sounzan Station (9 minutes). (runs every 15 ・30 minutes)
4) Hakone Ropeway - Sounzan to Owakudani Station (13 minutes) (runs about every 50 sec)
5) Hakone Ropeway - Owakudani Station to Togendai Station (22 minutes)
6) Lake Ashi - Togendai to Moto-Hakone (40 minutes). (runs every 20・0 min)
7)Tozan Bus - Moto-Hakone to Hakone Yumoto (40 min)
8) Romancecar - Hakone Yumoto to Shinjuku (17:49pm to 19:19pm)

By the way, must I buy the Freepass on the spot when I arrive the Shinjuku station or do I have to book the Romancecar ticket in advance since reservation of seats is required.

For the return trip, must we specify upfront the time to return when purchase the Freepass. What about if we miss the Romancecar that we suppose to take. How does this Romancecar system work.

Appreciate your kind advise. Thank you and great day ahead!

You can reserve romance car seats over the internet up to, I think, 2 months in advance. Go to the Odakyu Railroad web site. They'll need a cc number, but won't charge it until you pick up the tickets. You can even specify where you want to sit.
The first romance car in the morning leaves at 0730 which would give you more time, but 0730 is before their ticket office for tourists opens in the morning, so you'd have to pick up and pay for your reservations the day before.
Your schedule is obviously carefully worked out, but look out for the one possible fly in the ointment: fog. If it's a foggy day, the boats on Ashi no Ko can be cancelled until the weather clears. For that reason an earlier departure might be in order if you want to get back per your schedule. I'm not sure what you'd do if you miss your train in terms of costs, but the trains back to Shinjuku are rarely sold out.