I flew with both of my children (ages 13 mos. and 2 1/2 yrs. at the time) by myself from Germany to Baltimore I think it was? then on to Atlanta (and a 2 hour car drive after that)...and we all arrived fine albeit I think my make-up was a little 'off' afterwards.
I brought the big car seats with me both times flying and never had any problems. It REALLY helped that I had one of those double strollers because I just put their carseats inside the stroller itself and then they sat in those carseats instead of my trying to push a stroller and drag car seats, luggage, etc. around with me.
I always asked before each flight if they were going to be boarding families with young children first. Sometimes it seems like it was routine for them, sometimes you'd think I'd asked them to serve me caviar on a gold plate....

It REALLY helped to board first and I didn't have to feel bad possibly knocking the other passengers in the head with my big, clunky, car seats. Now granted, these flights were upwards of 10 hours long so no way was I going to hold either of them for that long. I liked having my hands free, etc.
I am curious though...I specifically remember not having a window seat at all on one flight (I *think* it was the international one, this was a few years ago) and, in fact, we were in a middle row of about 5 seats. (Pretty sure because of the size of the flight it was the international one.)
Anyways, just wondering if the rules changed recently or was it because I was on the Germany to USA flight that we didn't have to have window seat?
Okay, so here are my tips to help keep little ones occupied/comfortable:
In my carry-on I brought easy to carry snacks pre-packaged in the snack-size ziploc bags- things like Animal crackers, etc.
I bought the mini versions of several games including Etch-a-Sketch, etc. that you can find at
Walmart, among other places. I also bought cheapy stickers that they could stick on the seat in front of them. NOTE: I tried them out at home first (by myself) to make sure they would remove easily from plastic, etc. I CERTAINLY don't condone sticking them and leaving them on the seats! I later bought those toys that have the blank scenes and magnetic characters that you can move around to make your own scenes with and those were a hit! I saved all of the above as a surprise for the kids after we'd settled into the plane. (Of course, some cool Disney stickers would be perfect for them.

)
We were fortunate to have the tv's in the back of each seat for one of the flights so the kids could pick what they wanted to watch. When I booked the trip I do remember asking for a flight that had it and I managed to get one. Again, this might be only for overseas trips, etc. Another tip is to ask for bulk-head seats. It might to be too late to do that, but I didn't even know about it as an option until an agent there at the airport offered it to me. (This was during that 5 hour layover.) She came up to me and said how great my kids were behaving (again I credit some of that to the mini-toys I'd bought them!!

) and asked where in the plane were our seats...I showed her my tickets and she arranged to move us to the bulkheads. You are directly below a TV and behind a wall, but you have so much more legroom since there are no seats directly in front of you. Even during a part of the flight, my son got down on the floor and played and we had no problems!! Loved, loved, LOVED the bulk head seats. Now, granted the view to the TV wasn't all that great, but it didn't matter to us.
The only time my kids cried was during ascent/descent (sp?) of the plane...Mine never took to a pacifier and I nursed so I couldn't offer them a bottle and they, of course, are strapped in during those times. They obviously weren't old enough to chew gum, but your 6 year old can. It does seem to help with the plane pressurizing, etc. With my DD, I just had to bear through it and the crying stopped once we reached altitude. Besides, I don't think anyone could hear her over the sound of the engines, etc...she wasn't that loud.
*I'm sure your trip won't be near as long as mine was, but these things helped ALOT to keep my sanity. If I can fly by myself with two children for that long and still be able to say my ABC's after arriving, them I'm sure you'll do just fine.
Have a great trip!!