I'd agree that a weekday is probably better - as many Japanese will travel at weekends to Hakone to enjoy the countryside - particularly in autumn when the autumn colour will be great. The line also has many hydrangeas planted along it.

Hi Not sure of all opening hours but we managed to get train from tokyo to hakone then train to gora then cablecar to lake and boat across lake and public bus back to gora to stay overnight in one day with no great hurry and plenty of stops. Left tokyo approx 9am, back to gora approx 5pm. Everything closes early in gora (by 5.30pm) so i wouldnt expect late buses or trains in that area but there may be slightly more in the main centre of hakone.

1. No point driving to Yumoto; not just about parking but it is more difficult to get there than train. Park at Odawara or any other major train stations where parking is ample. Yumoto is a bustling place with lots of tourists almost anytime of the year. 2. Child ticket recommended unless u plan to put her on your lap on rides which are full 3. Hakone Free Pass is inclusive of train rides, Tozan Railway, Ropeway, Cruise and very well connected and little chance to get lost as very good signs and directions in English with staff stationed along the way to guide tourists. 4. Shinjuku, Odawara stations offer Hakone free pass or any major travel agents like JTB, HIS. Online is best though check if they mail it or you need to collect at a specific location nearest to you. Travel agents also can send the pass to your home or desired address. Travel light with your daughter as everything is available along the way; water, food etc though up on Hakone, temperature is usually lower than at the foot so a depending on the month of visit, bring a jacket. if child is not able to walk too much, then bring along a stroller which is easier and fast to fold up as will need to fold it up on Ropeway etc. Avoid weekends and public holidays if you can. always check the Public holidays in China (must avoid) as it will be packed with major groups....

You may want to consider the Hakone Free pass. See my review on this and there are a lot more details on planning and the pass as well journey time/details from Tokyo.