I've restored a metric ton of coin doors lately, and I can tell you that the door you have is in fine shape. No, really.
First, remove all the parts you've got, and completely disassemble the door. This includes the doors from the frame. Put all the parts into a container of some sort.
Get yourself some fine grade steel wool. Start scrubbing all the metal parts to get the corrosion off. Feel the parts with your fingertips to find the remaining corrosion.
Next, get yourself a scrub brush. Wash these parts with dish detergent and scrub well. Then rinse well, make sure to get all the detergent off.
Once the parts are dry, take some gold spray paint and LIGHTLY spray these pieces. I like to bake them in the oven at 200 degrees for about 1/2 hour. Let cool, finish off the other sides and repeat.
As for the door itself, get yourself some paint stripper. I bought a gallon, and it goes a long, long way. Coat the door liberally with it (depending on the stripper, read the directions) and wait for it to bubble off. Use a wire brush, and brush the coating off. You may have to repeat this.
You'll probably want to wash the remover off the door, as it's highly toxic. I don't know if you have a rotary tool, but using a polishing stone is great for removing rust. You can also use a wire wheel, or any other mechanical method of removing rust. You can use Naval Jelly, but I don't recommend it.
After you've removed the rust, scrub the door and frame with your brush and detergent, and wash until all traces of detergent are gone.
If you haven't already done so, get some Rust-oleum texture paint, and some Rust-oleum satin paint. Paint the backs first with the satin. LIGHT COATS. Wait a couple minutes, and give it another LIGHT coat. In the oven for 1/2 hour. Wait until cool, then repeat for the front, except use the texture paint. Do the parts SEPARATELY. Don't assemble them for a short cut, it'll look like crap.
For the sides of the door, you should also do them with the satin paint. If you do it with the texture, it'll build up, and you won't be able to open or close your door easily.
I note that your door has silver insert pieces. These are easy to do, as all they require are a good cleaning. You might want to follow up with a polishing wheel to restore shine.
As for the red coin rejects, I can't tell how bad they are as they're still in their housing. If they're not broken or badly scratched, scrub with an old toothbrush and detergent. If they are, those are available from Happs.
That's pretty much it; reassemble and you have a nice restored door.
If you have any questions, feel free to post here.