Hi vivian,
Is Hakata the same place as Fukuoka as used in your planning?
Yes - for JR, Hakata is the same as Fukuoka and is the main Bullet Train terminal for Fukuoka (Hakata is the old name for half of the current Fukuoka city).
Is the train fare from Osaka to Kansai airport 3140 yen and about 1, 300+?
The train from Kansai Airport to Osaka is ¥1,160 each way.
We are travelling on a tight budget and wonders if it would be more economical to buy Kansai Through Pass for two days at 3800 yen to use from Osaka to Kyoto and Kobe.
Yes, I think the Through pass will be good value considering the amount of travel you intend to make. It should cover transport from the airport into, and travel into and around Osaka, Kyoto and Kobe. Highly recommended. You can find more information on the Surutto Kansai pass online.
Would it be cheaper to buy two 7day JR Pass then one 14 day pass?
I don't think so. In your case, your long distance travel starts from the 8th all the way up to your date of departure towards the end of your trip, so you will use the JR pass pretty much in a contiguous block. Also on a per-day basis the 14 day rail pass is cheaper than the 7 day rail pass so 2x7 days will always cost more than 1x14 day pass:
7 day rail pass: ¥4042/day
14 day rail pass: ¥3221/day
You would be better off with a single 14 day I think.
Hope this helps!