I think you've got the right idea. Staying on Naoshima overnight was what we did when we went, and it really let us soak up the atmosphere at different times of the day (BTW staying in the fancy pants Benesse House is highly recommended if your budget gets that far). You should be able to get to Naoshima pretty quickly from Kyoto, so travelling out there in the morning should get you in by midday(ish).
So then, onto your questions!
Naoshima by Japan Rail Pass
"As far as I can see, the best way to get to Naoshima by Rail is to go by bullet train to Okayama, then change train to the local Uno line and go direct to Uno where we'd get on a ferry to go to Naoshima. I'm not sure if we have to change train at Chayamachi halfway?"
Yes, the best way to Naoshima using your Japan Rail Pass from Honshu is to aim to get the ferry at Uno. (If you're on Shikoku, there is also a similar ferry that runs from Takamatsu).
How to get a JR Pass?
You already found this site, so you're already half way there :) How to get a Japan rail pass is relatively simply. Go to the order page and fill in the details, we'll have the pass with you in a couple of days.
Kyoto > Okayama
As you've probably guessed, there are loads of bullet trains going from Kyoto > Osaka > Okayama, so you'll be able to get down there very simply. You'll have to change bullet at Osaka to a Railstar or Sakura bullet, but aside from that, the journey to Okayama should only take you about an hour. Here's a sample itinerary from Kyoto to Okayama by bullet train that is compatible with your Japan Rail Pass:
Okayama > Uno
There is a direct train from Okayama to Uno, but it doesn't run very frequently during the day, so you will be best to change train at Chayamachi. Fortunately, Okayama > Chayamachi > Uno route is pretty simple (and weirdly faster than the direct route). Here are some current timetables for the non-bullet bit of the journey between Okayama and Uno for comparison:
Uno to Naoshima Ferry
"Can I use the JR Pass on the ferry from Uno to Naoshima or is it a privately run company? Also, if it's not a JR ferry does anyone know how much the ferry tickets are? Do I have to book the ferry in advance or can I just turn up and board?"
Unfortunately, you can't use your Japan Rail Pass on the ferry between Uno and Naoshima - it's run by a private company. Luckily, it's not going to break the bank: adults are ¥280 one way. Here's a full translated pricelist and timetable for the Uno to Naoshima ferry. It looks like it takes 20mins or so on the boat. As you'll be getting on as a foot passenger, you can just turn up and buy a ticket when you board - so it's ultra simple.
Hope this helps and you have a great time on the island - I'm happy to answer any other questions you may have about it!