You really don't need to put it in a large plastic bag and it may not work to do so - you will be wheeling the thing up to the door of the plane where you will then fold it up and leave it resting (you must already have attached the gate check tag to the stroller you will get from the gate agents). People will be filing past you as you do this and you don't have a heck of a lot of room. I have not had an issue with the stroller getting harmed or ruined so I am not sure I would feel a need, even if I felt it was feasible, to place it in a plastic bag. If you do place it in a bag, I will bet, but I am not sure, that they will insist that it is clear so they can see through it. If your trays do not come off without a screwdriver, then no, you do not have to remove them. The baggage handlers simply set the strollers to the inside of the baggage compartment - they don't fling them into the back of the airplane and when you land they simply pull them out and set them back where you left them, right at the airplanes door area when you exit the plane and before you go down the gangway.
Gate checking strollers is the next best thing from slied bread. Its wonderful that the airlines allow this.