Yes, egg shells can also be a good source of calcium. A dish of eggshell should work about the same as a dish of oyster shell. The only problem is that you need a source of eggshells, because hens need more calcium than what they can get just from the shells of their own eggs. If you have to buy something, it's usually easier and cheaper to buy oyster shell than to buy eggshell, which is why oyster shell is more common.
Not correct.
I have watched hens lay eggs: a normal egg has a hard shell when it comes out, even if I pick it up immediately so the hen can never sit on it.
An egg that is laid with a soft shell or just a membrane will never develop a hard shell, no matter what happens after it is laid.