The only stroller that can do gate check is the umbrella stroller - the primitive ones. We used to have a Chicco stroller Ct 0.5 - that one was too big to do gate check in both CPH and ARN (was able to argue our way through at CPH a couple of times). You can get a tag for your stroller and you can put it on (they will ask you to put it at the odd size place), so if you need it after getting the tag, you could just bring it with you. There are stroller in CPH that you can borrow.
When you drop off your bags - you just need to show your boarding pass and the other pax don't have to show up.
I have done morning bag drop for a SK flight later in the day and did evening bag drop for a flight the morning after.