There are onsen, many options, in all the areas you are considering (just not on the slopes of Mount Fuji). But you are also traveling at a time when many will close for the holidays.
It’s up to you to decide what experience you are after - a neighborhood sento (quite a few will technically be onsen), a super sento Special Edition of "Super Sento" in Osaka! Recommended popular facilities chosen by Onsen Lovers!?EnjoyJapan! , and, most difficult to arrange with such short notice, is a stay at lodgings with a private onsen or/and rotenburo. Although there are many options, you might find that the date you decide to dedicate to onsen culture makes it more limiting/challenging than it would be at another time. Ask your hotel to check the bath house is open before you go.

Some time ago I kept a thread going with options in Tokyo, it’s out of date now and online map resources have invalidated the need for such a thread, but it will give you an idea as to the sheer scale of potential answers to your first question.
Tokyo Onsen

There is also the unasked question about where you can go if you, or your companions, have tattoos. That isn’t in my remit.

Second question I already answered. It takes about 30 minutes to commute from Osaka to Kyoto. Put a start point into Google Maps, put in an end point, select the date and approximate time of intended travel and it will give you the options and pricing for the route. Easy peasy. When searching for a start point using Google remember that Honmachi is also spelled Hommachi.
I just checked - 30 minutes from Osaka Station or Umeda Station to Kyoto Station, 40 minutes from Hommachi Station to Kyoto Station, 50 minutes from Sakaisuji Hommachi Station to Kyoto Station. The Moxy Hotel is a 5 min walk from Sakaisuji Hommachi station.

Third question. It depends. And everyone seems to muddle along using their own preferred methods. Credit cards are widely but not universally accepted. Cash is widely but not universally accepted. In between is an IC card, if you are starting in Haneda your initial choice is likely to be the SUICA transport card from JR, you can get this and use it in Kansai also. Most straightforward way to charge and pre-load an IC card is with cash. Once it has money in it you can use it all over for drinks and snacks, even meals, and not just for transport. I understand that taxis will accept IC cards also. I used a Wise card this summer with credit card as a back up and, along with cash and a SUICA card did fine.
If you have an iPhone (I don’t) you can get a virtual IC card, that will lessen your reliance on cash further.