Regarding the stroller:
Last October we flew with 2 kids from JFK to Rome via LHR with British Airways (got a good deal on WT+. In hindsight it wasn't a good move given the layover.....).
BA told me on the phone that as long as it is an umbrella style stroller that can fit in the overhead bins I could bring it on-board.
When we got to JFK they wanted us to check it in with all the other bags. We told them that we needed it for the transfer in LHR (especially since we have to change terminals) but they insisted and said we got wrong info over the phone. After arguing with them for a while they reluctantly let us take it up to the gate and check it in there and said that after that the next time we will see the stroller will be in Rome.
The ticket person and the station manager who came to talk to us were both unfriendly and did not try to help us at all.
After we were done with them and stood annoyed some distance from the BA counters another BA CSR came over and told us that she overheard the exchange and that if we make sure to be at the family pre-boarding call and ask nicely we might be able to get the stroller on board. That was indeed the case - the gate people and FA's on board were very helpful and friendly, played with our baby, ask if they can hold her, etc. Having the stroller in Heathrow was a life saver.
Once we got to the gate in LHR for our flight to Rome we had to go through same process again but they weren't too forceful - we told them we had no problem getting it into the plane in JFK and they did let us take it on board again.
We didn't have any issues coming back to JFK (again through LHR).

Bottom line - it is hard to predict what they will require from you to do. I wouldn't count on seeing the stroller until final destination if you check it in whether at the counter or the gate.....