let me answer to your questions one by one.
- Tokyo => Takayama
2 hour 8 mins from Tokyo to Toyama by JR Hokuriku Shinkansen.
1 hours 14 mins from Toyama to Hida Furukawa by Limited Express (JR Takayama Line).
:By Bus (Nohi Bus / Keio Bus)
5.5 hours from Shinjuku to Takayama.
15 mins train ride from Takayamato Hida Furukawa.
For 1. By train, I recommend "Hokuriku Arch Pass".
This allows you to use almost every trains including Shinkansen from Tokyo -> Toyama, Toyama -> Osaka during your trip for 7 days. All you need is just only this pass and will cost a very few coins to take some local buses or taxi when you need.
You can also take bus from Tokyo but I think it's too tough for old people according to your itinerary.
Seats are not so wide, takes more time... and, it seems you have a lot of places where you HAVE to walk around to see, especially in Kyoto.
I recommend you to save your energy as much as you can for Kyoto or Osaka.
If you want take the bus ride, please check https://www.highwaybus.com/gp/foreign/frgSelectBordingDate.
I think this is the cheapest.
Q2. No, I don't think so as I mentioned above.
Airplane or Shinkansen is better.
I don't recommend buses for the long distance rides, for old peoples.
Q3. Cheapest would be definitely buses only, I guess.
But I also think the Hokuriku Arch Pass for JR trains would be the best answer to cost you less than taking so many buses at last.
You can save your time, and you maybe able to save your transportation costs.
Q4. It's too far. Takayam -> Shirakawago 1 hour then Shirakawago -> Hirayu will 2 hours.
In stead, I recommend Hirase Onsen if you want to visit Onsen after Shirakawago, then get back Takayama in one day.
Takayam -> Shirakawago will take an hour by bus, and Shirakawago => Hirase will 20 mins.
The buses won't go/stop a lot.
So please check the timetable before you go.
EX) Leave Takayama at 7:50 ->Shirakawago at 8:40.
Leave Shirakawago at 15:02 -> Hirase at 15:18.
Leave Hirase at 19:24 ( this is the final bus bound for Takayama which stops at Hirayu) -> Takayama at 20:48.
NOTE) There're a few buses stops at Hirase. if you miss 1, then you have to wait for hours. Japanese transportation system is basically operated so much punctually.
The timetable for Takayama - Shirakawago - Hirase is;
-Takayama Green Hotel ,1,000 yen
It has foot bath for free as well.
-Garyuu no sato, 800 yen
They are the famous and very open for foreigners.
But it really worth to ask your concierge, there are real "Local public bath" in Takayama.
Q6. I searched Onsen in Kyoto, then I get so many answers.
20 public bathes are in Kyoto city at least and I'm sure there are more of them, about 100.
Where is the best place or worth to try depend on where you stay.
So, I think you should better to ask your concierge at the hotel you will stay.
You can ask about Osaka as well in the same time.
Hope this can be your help, and don't hesitate to ask more!