Need help with train trips

Hi there!

Here are some thoughts for you:

Matsumoto to Takayama

You are correct that the Nohi bus from Matsumoto to Takayama will not be covered by the JR Pass, so this will not be included.

Takayama to Koya san

From Takayama to Mt Koya, the JR Pass will get you a lot of the way there. For the final non-JR piece, you have 2 options I think:

1. Travel from Takayama to JR Nanba station in Osaka. From there transfer to the nearby Nankai Nanba station and catch the non-JR Nankai line upto Koyasan. (this is the shortest route, but uses the JR Pass until JR Osaka Nanba)
2. Travel from Takayama to Hashimoto in Wakayama which is significantly closer to Koyasan. From Hashimoto board the Nankai lines up to Koyasan. (this is the longest route, but uses the JR Pass until Hashimoto)

In your situation I think I'd opt for the Osaka Nanba option as it is the simplest route overall, but you may prefer taking the train to Hashimoto on JR as this will be the cheapest option.

Hiroshima to Tokyo

For this leg you must change Bullet trains at Kyoto as there is no through service unless you use the Nozomi trains which are not covered by the JR Pass at this time.

I hope this helps!