Hello,
So our itenerary is as follows:

Oct 8-12 Tokyo (4 nights)
Oct 12-18 Kyoto (6 nights, last day is travel)

We are also planning sidetrips, like possibly a sidetrip to Nara from Kyoto. I was really hoping to go to Hakone for a night between Tokyo and Kyoto but for the life of me, I cannot find a ryokan with availability on October 12th! Is it possible to do the trip to Hakone as a daytrip from Tokyo without staying overnight? Is it even worth it if we can't stay at a Ryokan? Are there any onsen we can go to without staying at a Ryokan?

Thanks!

Yes to all of the above.

There are tours available from Tokyo. Pickup and dropoff at your hotel (probably). You could also take the train yourself. A good way to go would be on the Odakyu train with a Hakone Free Pass:
http://www.odakyu.jp/english/freepass/index.html

It is worth the trip because it is scenic and you can get a break from the city.

In Hakone Yumoto there are ryokans that have their onsen baths open to the public.

http://www.japan-guide.com/e/e5209.html

I visited Tenzan in Hakone Yumoto.

If you look at the japan-guide map you can see the route I took: Yumoto (aka Hakone-Yumoto) to Moto-Hakone to Togendai to Owakudani to Gora to H-Y.

We took the train from Tokyo to Odawara at about 11 am. We then took a bus up to Togendai. Then we took a cable car, a tram and two trains back to Tokyo by about 7 pm. It was very easy and a lot of fun.

We just stayed a night at Lalaca in Gora, and it was very good. A modern building with a small but natural hot spring bath, nice rooms, and fantastic food. Location we loved, too. Very nice owners. They have rooms on October 12. www.lalaca.com

For info.
http://www.jnto.go.jp/eng/location/rtg/pdf/pg-307.pdf

Many ryokans/hotels give a day onsen bathing.
Try these ryokans/hotels w/english sites:You may need to adjust your trip to weekdays.
http://www.kanagawa-ryokan.com/chinese-s/hakone.htm

If all fails, try this. my value favorite. No outdoor onsen but view of the lake from room and nature surroundings good for morning walk.
http://www.hakone-lakehotel.com/engl...glishindex.htm

Thank you so much everyone!