Don't worry...............
I have the same oven and it's definitely a '57 (not 60's or 70's). It doesn't sound like a hinge is broken, rather a stuck return spring. None of the less, the oven has to come out of the wall to inspect the hinge channel. Please keep us updated on your progress.
I had a similar problem...........
But with my bottom door on my '56 Hotpoint combo. I shimmed it and shook it and bent the hinges and it's still not quite there but functional.
The contact clips...............
are going to be harder to find though. they do come up on Fleabay occasionally but you might have to get them from a donor oven. Are yours that bad ? Can you get by with just cleaning them off or are they destroyed ?
E Bay is your friend today............
You can find the same three prong plug-in element on Flea Bay. Just do a "Vintage GE Hotpoint" oven search. there are two sizes, a medium size (which I use for my vintage wall oven) and your size. Let me know how you make out.
Keeping it on the counter exposes the butter to varying temperatures. If it's kept in the controlled butter dish inside the refrigerator, the temp is more constant. It's really just a preference.
OK, here's what those "rods" are
The "rods' you are talking about are not two support rods but one C shaped bar that is attached to both ends of that shelf. I'll post a picture of my small shelf shortly.
Yes.....
That was in the top of the line model. It was the "Super Stor" feature, but you really didn't gain anything by it, you actually lost room in your fridge.