It should not be counted as one piece of hand carry. When you are at the boarding gate, let the FA know that you have a stroller. She will give you a tag to hang on the stroller. She will then tell you either leave the stroller with her now, or keep it until you are right outside the plane (pass the connection 'bridge'). If so then leave the stroller at the door right outside the plane and the staff will take the stroller and keep it for you. And when you arrive, don't forget to wait at the same spot (outside the plane) to claim your stroller.
If the stroller comes with the infant basket, depends on whether you purchased a seat for the baby or not. If not, then also leave the basket with the staff, unless the FA has a storage space to store the basket for you in the plane.
As other jms said, 間間航空公司唔同, you can double check with the staff when you check in. Most of them are very helpful and kind to the passengers with infant.
And yes, call the airline and ask for the bulkhead seat (first row) as soon as you booked the flight, so they can reserve and provide you a bassinet. The only disadvantage of the bulkhead seat, is the seat handle cannot be lifted up...
Also ,most of the time they will call 'passengers with infant' to go on board first, before they even call the first class passengers, so you will have enough time to settle down in the plane.
I am confused... when you were at check in they said '我冇幫bb買機票'.... so did you purchase a seat for BB? How much was the baby ticket? A full price ticket or just 10% of the ticket price + tax? Or you did not pay any for the BB at all?
Because even the infant does not occupy any seat (infant in arm/on lap) for international flight you still have to pay about 10% of the adult ticket price. So if you said you call to reserve bassinet they should aware that you have a baby in arm, and the CS should have also mentioned that you have to pay for baby's ticket (usually you don't pay until you are at check in counter...)
And the truth is bassinet is not guarantee nowadays even you called in advance... they rather give the bulkhead seatsto the passengers that pay extra for those seat, than the babies in need!
I am so sorry for your bad experiences... when are you going back to the States? You may want to call the airline to confirm that they know you are traveling with an infant!