Concerns about renting bicycle in Arashiyama

I would like to check if it make sense to rent a bicycle for the following 1 day itinerary in Arashiyama:
1. Monkey Park
2. Togetsukyo Bridge
3. Tenryuji temple
4. Bamboo Groves
5. Okochi-Sanso Villa
6. Jojakkoji temple
7. Gioji
8. Saga-Toriimoto Preserved Street
9. Otagi Nenbutsuji
10. Daikakuji

My concerns are:
- It may be hard to get to Monkey Park on bicycle, is there place to leave bicycle before climbing the mountain? Would it be safe?
- While visiting temples is there usually a place to park bicycle near the entrance?
- Is it allowed to ride bicycle in Bamboo Grove
- How steep the rest of the route, would be hard to ride bicycle from sight to sight?

Thank you very much.

It may be hard to get to Monkey Park on bicycle

It is impossible. You should park your bicycle at the base of the mountain.

is there place to leave bicycle before climbing the mountain?

I am pretty sure that there is some space available. Just be careful not to cause any inconvenience to anybody. If in doubt, ask the person at the ticket gate.

Would it be safe?

Yes.

While visiting temples is there usually a place to park bicycle near the entrance?

Yes. There are often a few parking lots around where you can put your bicycles.

Is it allowed to ride bicycle in Bamboo Grove

Yes, unfortunately even cars are allowed to pass. Of course, if it is very crowded, you better dismount.

How steep the rest of the route, would be hard to ride bicycle from sight to sight?

Only the slope to Okochi-Sanso Villa through the bamboo forest is steep. The rest is pretty flat and easy to cycle.