Maybe someone can post a pic of their ashtray so you can get a mental picture of what you're missing- then make a similar piece out of a scrap of thin sheet metal. Drill out the old rivets and pop rivet the new piece in place.
The original was spring steel, but a standard metal piece should suffice, since like you say, you're not a smoker and the thing isn't going to be opened and closed a hundred times a day. Regular sheet metal will also allow you to adjust (by slightly bending the tabs) the tension on the tray to ensure it's held firmly closed.
As far as your other question; yes, the U-shaped part is for holding the cigarette. The "grate thing" is for stubbing out your cigarette.
