One pass or two??

Hi,
It really depends on your itinerary. You need to be constantly moving about to make the 2nd pass worth it. If you are staying in one area, it may be better if you still want to travel, to get a regional pass in addition to the JR pass instead. For example, JR West has numerous passes depending on when and where you are going, as you can explore here. In answer to your other question, yes the days are counted consecutively.

You can see regular fares, routes and schedules on Hyperdia. But be sure under "More Options" to deselect "Nozomi..." since you can't ride that on the Pass. Just add up where you are going and compare it to the cost of a related pass.

For Fukuoka, there are many open parks, shopping, seaside historical areas, and urban neon. See this site as a good guide.
Don't miss the Nanzoin Temple which is one of Japan's finest yet almost totally unknown to the outside world and completely FREE!
Look into a Fukuoka Tourist City Pass also.
You can also see a lot of previews on the best there is to see here. If you want to travel around Kyushu, then there are two kinds of Kyushu Rail Passes for either 3 or 5 days that might help you too.

Best of luck.