Or I could take it in Takayama and go via Kamikochi -the dropoff charge is higher but I will save on a bus ticket to Kamikochi-Matsumoto (if the idea with hiking down to Kamikochi from Shin-Odaka onsen doesn't work out).

I can follow the "romantic road" for 2 or 3 days. What is the best place to stay overnight, the best places to stop, the best possible detours? Any not-to-miss suggestions?

Is it worth it to make a detour and go to see the place with monkeys in hot water near Nagano (forgot the name)?

If I go by car, I will not have time to go to Shirakawago - perhaps, there are other similar gasho villages on the way?

I saw the "romantic road" page in japan-guide but it doesn't have many detais.

I believe that by late October, the mountain ridge between Kamikochi and Shin-Hodaka will be covered by snow already and the hike impossible, but I am not 100% sure.

I can follow the "romantic road" for 2 or 3 days. What is the best place to stay overnight, the best places to stop, the best possible detours?

I recommend Kusatsu Onsen and Nikko as places to stay overnight, but if you see a place that attracts you more, I recommend to stay there.

Is it worth it to make a detour and go to see the place with monkeys in hot water near Nagano (forgot the name)?

Yes, most people who go there, enjoy it. So if you have the time, I recommend to make the detour.

If I go by car, I will not have time to go to Shirakawago - perhaps, there are other similar gasho villages on the way?

Takayama has an open air museum that is almost as good as the originals in Shirakawago:
http://www.japan-guide.com/e/e5901.html

We found S-go disappointing, while Gujo-hachiman was a real hidden gem really worth seeing!