Hello, welcome to FixYa.
I understand by your post you have a DCS RGS-305 in which the hinge "popped" out, correct ?
I'll address the removal/replacment of the oven door first. These doors are only meant to be taken off with the hinge "locked", as shown here.
1) Fully open the oven door and locate hinge lock lever. After locating the hinge lock lever, pull it to the lock position as shown.
example: Pull the oven door up at roughly 45°, and pull the oven door out of its hinge receptacles and away from the unit. That should address the oven door question. These are meant to come off in a certain manner as I mentioned so if the hinges are not "locked" when the door was pulled off, they will need to be locked before going back on.
Let me know if you have any trouble at that point and for sure I'll assist you. Good luck. If this indeed solves your plroblem please take a moment to rate the assistance here.
If it does not, please let me know and we can go from wherever you're stuck.
Regards,
Macmarkus :)