Koyasan & Kurokawa Onsen

Hi there!

We are planning our trip to Japan and we bought a 14 days JR pass.
Our planning is as follows:
6 – 8 April Tokyo
8 – 13 April Kyoto
13 – 15 April Osaka (1 day Koyasan)
15 – 17 April Kobe
17 – 19 April Hiroshima & Miyajima
19 – 20 April Fukuoka
20 – 21 April Kurokawa Onsen
21 – 24 April Fukuoka

1) We want to activate our JR pass from 8 April until 21 April.
Is our 14 days JR Pass also valid on the 21th of April?

2) We want to do a daytrip from Osaka to Koyasan, does the JR Pass cover this journey?
If not, what's the best way to get Koyasan and how much is a return ticket?
What's the last train / bus back to Osaka?

3) From Fukuoka we will head to Kurokawa Onsen. How to get there by JR pass? Do we need to travel via Mt. Aso Station or is it recommended to take a direct bus from Hakata Station? Is this bus also covered by JR?

Many thanks!