Help with travel itinerary - do I need a JR pass?

Hello,

My partner and I are traveling to Japan this June from June 3rd - June 24th for the first time! Our current travel itinerary is as such, but it is subject to change:

June 7th (Day 4): Travel from Tokyo to Nikko
June 9th (Day 6): Travel from Nikko to Nagoya
June 10th (Day 7): Travel from Nagoya to Kyoto
- Potentially do a day trip from Kyoto to Nara during this period June 14th (Day 11): Travel from Kyoto to Osaka
June 16th (Day 13): Travel from Osaka to Wakayama
June 19th (Day 16): Travel from Wakayama to Osaka
- Potentially do a day trip from Osaka to Kobe during this period June 21st (Day 18): Travel from Osaka to Tokyo

I have done the JR calculator pass, and it tells me that I still require another trip to earn the 7-day pass, however, since we are there for a longer period and staying in cities during longer periods, 7 days doesn't make sense for us. However, I get confused about the pricing whenever I look up the prices of shinkansen/trains in Japan as some state a one-way trip for 120 EUR (19000 JPY). Additionally, if I end up not needing the JR pass, is it possible to book the train tickets ahead of time and where could I do that?

Thank you for your help!