Tokyo/Hakone/Kawaguchiko/Kyoto

Wondering what the best way to travel between Hakone and Kawaguchiko and Kyoto is?
We will have two young kids and suitcases!
We'd like to visit the Fuji Safari park after Hakone (on way to Kawaguchiko) - is it best to hire a car? If so, where would be the best place to hire it?
How many nights would you recommend for Hakone + Kawaguchiko?
Taking into account our suitcases, and our desire to travel to Kawaguchiko afterwards, where would be the best area for us to stay in Hakone? ( I can't picture us taking all our luggage on the ropeways etc but maybe that's what people do?)
Lastly, would it be best to swap the locations? Do Tokyo -Kawaguchiko-Hakone-Kyoto?
Many many thanks.

Public transportation to Fuji Safari park is bad. Furthermore, exploring the safari park with your own car is - in my opinion - more exciting. So, yes, I would consider renting a car. And since you rent a car, you could consider using it during your entire trip around Hakone and Kawaguchiko. That would also solve your luggage problem. But try to avoid Hakone on weekends and holidays, because traffic get horrible.

Driving out of Tokyo is not very pleasant, though. So, for comfort, I would take the shinkansen to Odawara and rent a car from there. At the end, I would drop off the car off at Mishima or Shin-Fuji Station and take the shinkansen from Kyoto from there. It may also be cheaper to pick up the car in Mishima rather than Odawara because you could get rid of a possible drop-off fee.

How many nights would you recommend for Hakone + Kawaguchiko?

Entirely depends on your preference. But at least one overnight stay in total.

Lastly, would it be best to swap the locations? Do Tokyo -Kawaguchiko-Hakone-Kyoto?

It doesn't make too much of a difference if you use a car. If you were to travel by public transportation only, then the latter route would be slightly more efficient.

Thanks so much for your reply. That�fs really helpful.
Where would you recommend staying in hakone so we can still take advantage of all the fun transportation methods?
Is it worth doing a loop around the base of Mt Fuji on the way back to the train station/car drop? Thanks

Where would you recommend staying in hakone so we can still take advantage of all the fun transportation methods?

Anywhere along the loop would be fine. When searching for a lodging, I recommend to check their parking options.

Is it worth doing a loop around the base of Mt Fuji on the way back to the train station/car drop?

From Kawaguchiko, it is not that much longer to loop around the mountain via Fujinomiya and drop off the car at Shin-Fuji compared to dropping it off at Mishima. So, if you have a little additional time, and there are sites of interest on the northwestern and western side of the mountain, that could be an attractive option.

I would just pick up and drop of the car a Mishima. It is roughly in the middle of all the areas you wanna visit.

You'll probably need at least 2 nights minimum, but 3 nights would be better since you have travel to/from Tokyo and Kyoto.