Hakone itinarary

My family is planning a trip to Hakone with overnight stay in a ryokan at Hotel Okada. We have bought the package from Odakyu that includes Hakone freepass + Romance car tickets. We are taking the morning 8am train from Shinjuku and return the next day on the 3pm train.

I am thinking of the following itinerary:
First day -- reach Hakone Yumoto around 11am. Check into hotel. Then, go sight seeing in the Hakone area using the free transport on Hakone freepass. Finish the day around 6pm, return to hotel, enjoy the onsen before dinner in hotel.
Second day -- perhaps visit Gotemba for some shopping then return to Shinjuku.

1) Can I finish the important Hakone sights in about half a day on the first day? If so, which are the key sights that we must visit? I am traveling with spouse and 2 kids.
2) It seems traveling to Gotemba takes a bit of time and quite troublesome. Wondering if it is worth the effort and time.
3) It is the first time we are doing an onsen. Will it be too cold to enjoy? We are visiting in mid Dec.

Thanks for any advise or other suggestions.

Finish the day around 6pm, return to hotel, enjoy the onsen before dinner in hotel.

6pm sounds a little late. Sunset is around 4pm and there is not much to do after sunset. Furthermore, ryokan often serve dinner already around 6-7pm. So, I personally would try to be back at the ryokan around 5pm.

Can I finish the important Hakone sights in about half a day on the first day?

I don't know what sights are important to you.

If so, which are the key sights that we must visit?

There are none. Hakone is an overall experience. There is no single attraction that must be seen. I recommend to pick sights based on personal interest:
http://www.japan-guide.com/e/e5200.html

It seems traveling to Gotemba takes a bit of time and quite troublesome. Wondering if it is worth the effort and time.

Getting there is not too bad. But getting from Gotemba back to Shinjuku via Yumoto (in order to catch the Romance Car) is rather time-consuming. But I cannot judge whether it is worthwhile for you or not.

It is the first time we are doing an onsen. Will it be too cold to enjoy?

No. There is no better time to enjoy onsen than in the middle of winter.

Thks Uji.

Sounds like better give Gotemba a miss given the short amt of time I have.

Ha ha, I also feel enjoying onsen in winter should be a great experience ...... need to go convince my wifey .... :-)