Hey, I just went through this yesterday.
I kept asking who needs to look at my trade-in and they kept telling me that the next person I deal with will handle it. Well....nobody did. I was shown my Model 3 and thanked for my business and told to have a nice day. I asked them about forgetting to take my trade-in. They asked me if I had applied for a trade-in, etc. I told them yes and it was reflected in my paperwork. They took my paperwork inside and brought someone else out who took my plates off for me, gave it a quick once-over and then I signed over my registration. He handed me back my plate registration with my plates.
They literally forgot to take my trade-in. I wonder how long it would have been for them to figure it out. I almost forgot at the end due to the excitement but I put my hand in my pocket and felt my keys.
To answer your questions specifically:
-No, no clear spot to park. I parked to the right of Hall 6 in the lot by the loading bays.
-I did not trade those in but if they are in your car, you're fine.
-They took my keys, walked over, removed my plates, looked at the car quickly and then came back
-I think they just make sure the car isn't mangled or anything since they didn't do much of an inspection of mine.
Hope that helps!