I've done it several times.

The historic sites are close to the train station and many hotels. Hida no Sato (folk village) is a boring walk - take the bus. https://www.japan-guide.com/e/e5901.html#section_get_there

Takayama is better visited on foot especially in the old town.

take a look.

https://www.japan-guide.com/e/e5903.html

As Roger said in #2 summarizes everything.

The hiking trails start from public transport accessible places such as the Hida Folk Village which might require a simple bus ride.

Of course, if you are very interested in hiking, consider Kamikochi.

Why would you need a car? If going to the limestone caves out of the city, maybe, but most of the tourist areas are walking distance 10-15 minutes.

While Takayama is a city of 90,000 people, most people never get outside of the tourist area which is pretty compact.