1. is it better if we schedule this based in Tokyo or leave on our last day from Tokyo to Kyoto, with an overnight at Hakone? I am mostly worried about the fact that in the 2nd scenario we will have our luggage on our trip to Hakone (and we should check in first, meaning that we have to spend extra time to do that). Having that said maybe we shoud skip the round course and make it a full day trip from Tokyo?
For your luggage there are at least two possible solutions:
1) there is a luggage forwarding service within Hakone: https://www.hakone-tozanbus.co.jp/english/carry/
2) Or you could send your luggage from Tokyo directly to Kyoto and travel only with a minimal backpack for the one night.
Whether Hakone is better as a daytrip or overnight depends. I personally am a slow traveler and enjoyed my 2 nights stay in Hakone. Daytrip has the advantage that you are more flexible when to go, ie you can choose a day with good weather. Overnight gives you more time to explore but unless you are a last minute guy your travel days would be fixed upfront. From a geographic perspective going as a stop between Tokyo and Kyoto makes sense. 2. since we are mostly interested in the Fuji mntn views and of course the hot springs/hot baths staff (onsen experience), where is it better for us to stay? i noticed that there are several villages nearby but i guess some of them are not easy to access when you use public transportation.
Seeing or not Mt Fuji is based on the weather. There is ABSOLUTELY NO GUARANTEE that you can see it from Hakone. If Mt Fuji up close is the most important thing of this daytrip I would consider Kawaguchiko instead. Although even from there on a foggy day or just a day with a lot of low hanging clouds, no Fuji to see.
Regarding where to stay, if you want Fuji views from your hotel, I am pretty sure that hotels that offer that will heavily publicize that fact. In general places in Hakone are relatively easy to reach with public transport. Your chosen hotel will describe in the internet how to reach them. 3. buying the JR Pass, apart from the Hakone pass should offer us some extra discounts, right?
you should not buy the nationwide JR pass because of Hakone. Buy it only if it makes sense for the overall trip.
There are two “normal “ ways how to get to Hakone: https://www.japan-guide.com/e/e5206.html
If you have a jr pass or if Shinjuku isn’t a convenient departure place for you out of Tokyo, take Shinkansen or normal JR train to Odawara and change only there to Odakyu.
If Shinjuku is convenient and you don’t have JR pass, then most likely one of the Odakyu passes for Hakone will pay off. Just check it out if you should buy Shinjuku-Hakone/Odawara out of the pocket and get a pass only for Hakone proper if you aren’t going to come back to Shinjuku. I think that the normal pass from Shinjuku includes a return to Shinjuku so if you don’t , it is likely too expensive. But you can still get the pass for Hakone only.
Enjoy your trip to Japan!