Hello Lilly,
Almost your entire itinerary is covered by the JR Pass. Only the bus from Mt Aso - Kurakawa Onsen and transport (ferry) to Yukushima are not included.
Now, let's dig into your other questions:
Which JR pass should we purchase? We're travelling on a budget! So hoping to get the most out of our pass
I would recommend the 14 day JR Pass for the above itinerary. This is by far the best choice, there's a lot of expensive Shinkansen traveled involved and you can cover all of this if you use the pass starting from the 15th until the 28th.
Do the orders of Cities make sense?
Yes, the look at the map. The cities are arranged in a good way and there is no unnecessary extra travel time.
What is the cheapest (and efficient) way to get to Yakushima Island?
Here is a breakdown of possible options.
And any other recommendations! I am particularly interested in hearing your thoughts on great hiking and outdoor activities in Kyushu.
I certainly have a couple tips for you. The first is a short one but be sure to swing by Miyajima island when you are in Hiroshima. The Tori gate in the sea is a wonderful view and very iconic. Also good to know is that the JR Pass covers the ferry ride to Miyajima island.
On to Hiking, Japan is great for hiking and many trails can be found outside of the bigger cities. One great places for several good hikes is Mt.Aso. You can basically make a long hike around the volcano crater, as well as several other hikes in the area. In fact hiking is the main reason to visit Mt.Aso in my option. An other smaller but nevertheless good hike can be found near Kagoshima. From Kagoshima city you can walk up the nearby mountain range and go up to Shiroyama Park and have a beautiful view of Sakura Jima (the volcano in the bay of Kagoshima). An other amazing hiking place not far from Kagoshima is near Kirishima, this is a little like Aso though in that it is a volcanic landscape. Lastly there are some nice hiking trails in Kyoto as well. One is to hike up the Daimonji mountain. It's only a 40min hike but the view is well worth it, as you can view the entire skyline of Kyoto. A 2nd more secret hike is in the north of Kyoto is near Kurama-dera. This is a mountainous forest area with many small temples along the trail. My tip is to start the hike relatively late. Around 5 p.m. or so, this because the main temple closes around this time and there won't be anybody around. You can walk the entire train passing many small temples but not a single person. It's almost magical.
Any other recs on the below itinerary?
Yes! Every city in Japan has it's own (locally) famous food. This alone is worth the trip in many cases. My advise is to ask locally what the best dish in the area is and try that. This way you will experience Japanese food not found anywhere else!
Hope this helps!
Daniel