******Updated: I have never had the carseat and stroller count against luggage allowance, but airlines are changing policies so it is good to check. The last time I flew with a carseat and stroller was on United in March. Your chances of getting the bulkhead with a 10 month old are very slim, but it doesn't hurt to ask. Younger babies have priority, but first come first serve plays a role as well, depending on the person;s mood.
I would check the carseat. You can use the stroller and gate check it so they bring it to you for the layover. When I flew from Cairo through Frankfurt they left my stroller in Frankfurt multiple times, but at least you will have it for the long layover. I like to put my baby in a carrier like the Beco or Ergo and use the stroller to push the carry-ons. Of course, during a 6 hour layover, she may need to be in the stroller.
Whether you check your carseat or not depends on your child. At 10 months old, my son would not sit in the carseat on the plane. If your child is active, check it. You will appreciate the extra room to lay down with her or just let her play. If your child is fairly inactive and enjoys sitting and looking and doesn't try to move much, then maybe taking it is best. At 10 months, my son would have screamed in his carseat so we were much happier without it.
If you really want to bring your carseat, there is an attachment you can buy called the GoGo Kidz Travelmate. It is sold through http://www.onestepahead.com among other places. It attaches to you carseat and turns it into a wheeled bag. You can put your child in it and push her through the airport. If you do that, then I suggest you check the stroller.
http://www.onestepahead.com/catalog/product.jsp?productId...
If your child is still nursing, nurse her on the way up and after you reach altitude for a few minutes to help her ears adjust. If not, use a pacifier or a bottle. There are special rules for bringing on formula and pumped breastmilk. Probably not a problem going to the States, but review the rules so you don't have a problem when you are going back to Kuwait. Bring a quart sized ziplock bag (required for security in the US, put the stuff in on your way to the States so it is ready to go back for your trip home. If it isn't in a quart sized ziplock they will throw it out and you won't get it through security on your way out) with baby tylenol, teething tablets, etc. There are prescription ear numbing drops that you can get from a doctor that are usually given to kids with ear infections. You could get those just in case she has a really bad problem with her ears. If you nurse on the way up and down, she probably will not have a problem.
Bring plenty of snacks and a sippy cup for drinks. Coming from Kuwait you can probably have it filled going through security, in Germany and the US you can't. However, you can have the flight attendants fill it with water or juice when you board or bring pre-packaged juice boxes.
Bring more than enough diapers and wipes in case there is a problem and you get stuck in Frankfurt. They don't sell diapers in airports.
Bring some small distractions and a change of clothes (or two). If you have noise-making toys, make sure they are fairly quiet so they won't be so annoying to surrounding passengers. Have a blanket or some layered clothing so you can keep yourself and the baby comfortable no matter what the temperature on the plane or in the airport. Try to limit your carry ons to what you can manage by yourself. You can sometimes get people or the airlines to help, but it is better to be able to manage yourself.
If the airline allows passengers with small children to pre-board, then pre-board. However, ESPECIALLY WHEN TRAVELING FROM THE MIDDLE EAST: GET OFF LAST! Don't even try to get off until most of the people are off. People push grab their bags before the plane has stopped and push and shove their way out. With all the stuff you have to manage, it is better to wait. You have a 6 hour layover anyway so there is no hurry.
The last thing I would say is adjust your expectations and attitude from your pre-baby days. If you know that it is going to take you longer to get through the airport and that you will probably not get much rest on the flight, you will be less stressed out. Just take your time and it won't be as bad as you think!
Good luck! I've traveled internationally with my now, 2 year old son a lot since birth. Our next trip is in December. It is rough, especially by yourself, but you can do it.