If you want to see Hakone and Kawaguchiko, I recommend to allocate two nights. Otherwise, you don't have enough time.The recommended route would be as follows:
Osaka - Odawara - Hakone - Gotemba - Kawaguchiko - Tokyo
Alternatively, I recommend to drop either Hakone or Kawaguchiko. If the sight of Mount Fuji is most important to you, I recommend to drop Hakone and visit only Gotemba and Kawaguchiko, because the chance of seeing the mountain is higher from Kawaguchiko than from Hakone.
And is it advisable to climb Mt Fuji on 6th Oct when I reach Mt Fuji?
To the summit? No. It is not advisable to climb Mount Fuji outside the official climbing season. Furthermore, you have no time to do so. It takes about 24 hours.
I heard my sister talking about Oshin-hakkai village. She stayed there when she visited Japan last year. Will that be along the way?
Yes, it can be visited between Gotemba and Kawaguchiko.
So my suggested route for a 1-night stay is:
Osaka - (direct hikari train) - Mishima - (direct bus or trains via Numazu) - Gotemba (with shuttle bus ride to Gotemba Outlets) - (bus) - Oshino Hakkai - (bus) - Kawaguchiko - (bus or bus/train) - Tokyo
I recommend to leave Osaka very early in the morning. Three buses per day go directly from Gotemba to Oshino Hakkai.
If there is no time for Oshino Hakkai on the first day, you can relatively easily visit it from Kawaguchiko on day 2.
