I can help you a bit with Osaka, but not specifically about the JL 787 as I haven't flown it.
Here's a really good site to look at for train times and cost in Japan:
http://www.jorudan.co.jp/english/
They also have an app for iPhone or Android but it's Japanese only:
乗換案内 (You want the one published by Jorudan Co., Ltd). I believe the website is still functional enough on a mobile device, but if you want to use the app, entry in standard alphabet works and it'll pick up your intended stations if entered correctly, you can compare to your guidebook for Japanese characters.
From KIX (called Kansai Airport if you're entering it on Jorudan or 関西空港 on the app) to downtown Osaka will take 45 minutes to an hour depending on which train line you use (there are 2 operating the same route, neither is terribly fast). Regardless of what you do during the day, unless you're really itching to spend a long time in the lounge (it's nothing special as you've already uncovered in your research) then I personally wouldn't bother getting back to the airport until 1 hour before flight time, Japanese security and immigration is extremely efficient and rarely takes very long outside of peak holiday dates. If you want extra padding go for 90 minutes, but no reason for more.
As for Osaka, your friend is more or less right, it's not a terribly exciting city for a tourist. It's a shame too you're really arriving too early to do much until 9 or 10am so it cuts a bit into your practical time even though you have 11 hours. For food, the local specialty is called Kushiage. Essentially that's deep fried stuff on a stick. Mostly various kinds of meat but also fish, vegetables, and even ice cream. Any kushiage restaurant that's equipped for tourists will have an English menu but most likely the staff will not speak any English, be prepared to point to the menu and they'll write the order in Japanese. Also, no double dipping in the sauce bucket on the table, that's a serious taboo because it's shared by all diners throughout the day. So that'll pass an hour of your day, you could pass another hour going to the castle and some more time people watching and browsing shops in the Namba and Nipponbashi areas (they are next to each other). The more interesting nightlife though doesn't pick up until after 7pm where you can watch salarymen drink themselves silly.
If you want to go to Kyoto, it's certainly possible but plan out what you want to do as it's still quite a large and spread out city, and
definitely plan your return time and don't miss it. There is a direct train service called Haruka(はるか) (It's JR and requires a special ticket, so go to the JR office at the airport train station, not the Nankai line office which is the competing private line) that runs about every half hour during the time you'll be there, it's not terribly expensive (about 3,000 yen each way) and takes about 75 minutes. It's a bit under 90 minutes on the regular trains and under 2,000 yen that way but you'll be changing train lines once or twice that way and would be far more confusing if you're not familiar with the Japanese train system. Using Haruka you could easily be in Kyoto station by 9:00am (your first available train is either 7:27am or 7:41am depending on weekday or weekend, both arrive the same time though) and leave back to the airport at 2:15pm-2:45pm (again depends on weekday or weekend, those arrive at airport station at 3:34pm or 4:04pm respectively). That gives you enough time to see a few well planned sights there and it's a far more interesting city for sightseeing. If you're willing to spend $50 for what is going to amount to 5 hours in Kyoto is up to you, but I have a feeling if you decide to stay in Osaka all day you'll wish you would have, plus you'll still spend $15 round trip to downtown Osaka and back.
Whichever you do, don't worry about rush hour if using trains, but don't carry your luggage (it's crowded... very crowded), there are plenty of coin lockers in the airport to store your carry on bags, a really large one will only cost you 500-1000 yen for a whole day and you pay when you pick it up on the modern ones (which I think all at the airport are, but could be some old style too). If you're boarding the Haruka to Kyoto though I'd advise paying a couple bucks extra for a reserved seat so you don't risk standing up in an unreserved car. During rush hour in downtown on the regular trains, you'll get to witness a very unique part of Japanese life, everyone crowds into the train to a point you can't move your arms. Its unfortunately necessary because if you try to be polite and wait then the next train is just as crowded when it comes. Any ideas of personal space is right out the window during peak times (though Japanese don't respect personal space on trains or when walking anyway, just something you get used to).