We have traveled with this "Jeep" reclining umbrella stroller. It's been used for both of our grandkids and it was great. It reclines, has a shade and worked for the kids up until two. The kids had no problem napping in it for hours.
I believe you can still find it at Wal-Mart. It is listed online.
It is comfortable and MUCH easier to handle on planes than the regular stroller.
In regards to the car-seat. You may wish to bring it. Most cabs don't have seatbelts.