3 Questions to Save our Japan Trip

Hello! I’ve got multiple questions about the JR pass and when and how it works, as the rules where it shouldn’t work kinda confuse me... And I need to know the exact amount of money me and my friends are going to spend on our Japan trip in total before visiting the country itself. I hope asking this much questions is okay because I really wasn’t able to figure it out myself or by internet.

Question 1:
As we are staying in Tokyo for a week and I have heard that the JR pass is great to travel around there by train and bus, how do I know for certain which trains and busses are from the JR company and we can use the pass on, and on which ones we can’t? When looking for travel advises within Tokyo on Google Maps you get a lot of different icons next to the different busses and trains and I think I want to know which ones we can travel with, or maybe if there is an app/site we can use in English which can filter out all the ones who don’t use JR pass. I think it will save us a lot of hassle that way by not accidentally stepping in a train or buss that isn’t compatible.

Question 2:
After Tokyo we are planning going to Shimoda in Shizuoka, but from what I’m getting is that the furthest we can travel to Shimoda with JR pass is Atami station(? Please correct me if I’m wrong). Is it easy to step over from JR-train to non-JR-train like that? I know It’s gonna cost something, but I don’t know if It’s easy. Also please confirm if there really are no other ways with JR pass to go to Shimoda, that would help out alot.

Question 3:
When we are done in Shimoda we want to head over to Osaka by going back to Atami station and taking the train all way to Osaka. But I’m seeing a few sources saying you can’t actually take the train from there to Osaka with JR pass and I am very confused why. In addition I’d like to know if traveling from Osaka to Nara, and then from Nara to Kyoto is all do-able with the JR pass as well.

Thank you so much for reading this all and I hope to receive an answer soon~