Trip to Hakone

Hello, the family of four, including 2 girls age 7 & 10, will be in Tokyo from 20-27 Nov. We have been to Tokyo before and hence I am looking into visiting Hakone on one of the day. From my findings, the best way to go seems to be by train from Tokyo to Tokyo Yumuto (1 hr 45 min - Yen1750). When we are there, are we able to travel freely and easily if I intend to take the cablecare from Sounzan? How long is the journay for the cablecar and will we feel cold inside the cablecar since it will be winter by then. Another worry is that if we are on foot tour, will the cold weather be a problem since we are from Singapore which has a tropical climate. Your advise is very much apprecciated! Thank you.

From my findings, the best way to go seems to be by train from Tokyo to Tokyo Yumuto (1 hr 45 min - Yen1750).

Your options are explained here:
http://www.japan-guide.com/e/e5206.html

When we are there, are we able to travel freely and easily if I intend to take the cablecare from Sounzan?

Not sure whether I understand your question correctly, but yes, you are allowed to travel on your own without a tour guide. You will have to take a train from Yumoto to Gora and then a cablecar from Gora to Sounzan. The cablecar from Sounzan is known as "ropeway":
http://www.japan-guide.com/e/e5206.html

How long is the journay for the cablecar and will we feel cold inside the cablecar since it will be winter by then.

They are probably heated. In any case, it is not going to be colder in the cablecar than it is going to be outdoors. If you bring appropriate clothes, it should be no problem. November is not so cold yet.