I have taken this flight from Houston to Doha connecting in Dulles. I have 4 kids, including small ones, so I was watching it carefully in terms of how easy it would be with little ones.

Departing from the US: I checked my bags straight throgh to Doha, I did not have to re-check them in at Washington.

Departing from Doha: This flight is not very good, because it arrives too late to make any connecting flight. So I fly through Europe out of Doha.

You can bring your car seat on the plane, dont worry about that, and you will be thankful that you did, since kids tend to realize they arent going to move, and you wont have to wrestle with he for 14 hours. You can also try to stuff some blankets under the seat, and put it in a recling position, hopefull they will get some shut eye.

You will have to recheck your bags when you arrive in DC. Another reason why I plan to avoid that flight when we fly to the US. I still dont know why the US is the most annoying country to travel to, somethings about the good ole USA are as senseless as stuff you would see in other 3rd world countries. QA WILL CHECK YOUR BAGS THROUGH TO YOUR CONNECTION, BUT TRUST ME, THEY WONT GET THERE. Dont listen to the ingorant check in folks in Doha, they still have no clue what they are doing. They always check my bags through, and I always find them on the carousel at baggage claim.