L.D.
I know a lady who traveled to Turkey I think it was with her twin baby girls and husband. I think you are stressing yourself for no reason. You'll be fine, especially with the extra help you will have. :)
1) Take a pack and play. You can check it in as it is or you might be able to get a travel bag for it at Babies R Us or the like. When the lady above went to Turkey, she took 2 pack and plays so the babies had some place safe to sleep and play. She said it wasn't a problem at all.
2) You can get a travel bag for the car seat that allows you to carry it on your back like a backpack. I would take the larger seat so your child is comfortable.
3) Don't do an umbrella stroller. I swear those things should be banned. I highly suggest the Combi light weight stroller. It folds into thirds making it almost as small as a folded umbrella stroller, comes with a strap that you can sling it over your shoulder, but won't tip like an umbrella can, supports your baby's back, and reclines should your baby fall asleep. You may want to get a little stroller bag to hang from the handles though as the basket is very small and there aren't any cup holders but it is still a GREAT stroller for traveling. It would also be great to feed your baby in at the place you are staying (if you do solids) since I am guessing there won't be a high chair there.
4) Just don't forget some things for on the place. Even though your little one is young, I'm sure it will be a long flight so you want a few distractions to offer like a few stuffed toys so you can offer one and then later offer a different one to change it up. You could also bring some picture books, rattles, teethers, etc. Don't forget to take things like Tylenol, refrigerated teethers for when you get there, diaper rash ointment, a hat since the baby is still too young for sunscreen, etc. You can always get these things there but it's better to be prepared especially if you need any of the above while on the plane.
Enjoy your trip!!