@Bedelman: Thanks for this information. This is really valuable and should help me out for the outbound for sure, as we do have the new transit visa for the layover.
EDIT: I just signed up for the free trial on EF and the fare rules state:
"IF THE STOPOVER IN DOHA BETWEEN CONNECTING
FLIGHTS IS LESS THAN 24 HOURS PLEASE NOTE THAT
THE BAGS WILL ALWAYS BE CHECKED THROUGH TO THE
FINAL DESTINATION.THE PASSENGERS SHOULD CARRY
EVERYTHING THEY NEED IN DOHA IN A HAND BAG."
So anyway I will try my luck using the new fare rules as an argument.

@worldlux: You can gate check the stroller, but QR does not deliver it to the aircraft door or carousel in Doha unfortunatly, as this is usually standard for all other airlines. Eventough they often do not really know, where exactly it will be delivered to at the place of arrival.

@EvilDoctorK: We will have a sling carrier for sure as a backup. And I also checked with the hotel and they could provide us with a stroller in case we can't get ours.

Thanks to all of you for your help and input, this is highly appreciated.