is the 14 day pass a good deal for our schedule

Hi Wally!

Let's have a look at ticket prices and find out! Before we begin though, do you really mean Fukura (in Tophoku) to Hagi (Yamaguchi-ken?) This is almost impossible to do in single day. I only found one route which takes the whole day. You can find it here. I would honestly recommend looking into alternative transportation such as flying if you want to do it in a single day. Never the less I have included it in below

As for Toyooka, do you mean the town near Kobe/Hyugo or near Shizuoka? I've used to one near Kobe as it is the shorter route

Day 1 Fukura to Hagi ¥ 30,670
Day 3 Hagi to Toyooka ¥ 14,670
Day 4 Toyooka to Kyoto ¥ 7,530
Day 8 Kyoto to Niigata ¥ 21,080
Day 14 Yamagata to Nikko ¥ 9,140

For a total of ¥ 83,090 or ¥ 37,990 as the 14 day JR Pass is 45,100. So you would be making very very very good savings with the JR Pass however there's a lot of travel involved so I would recommend looking at changing your planning. For instance moving Kyoto to day would would spread travel time around in a better way.

Hope this helps!
Daniel