I understand that currency over $5000 must be declared. I want to take, say $50,000. Is this allowed? Is there a fee?

it is too much and it is not allowed.
It is suggested to take 5000US$ for cash and other money in your bank card.

OK, so how much is allowed? If over $5,000 must be declared, it doesn't mean you CAN NOT take more, right?

"If you have more than this amount, related documents will be needed and you need to declare to the customs." What are the related documents?

To be honest, I did not meet such kind of situation.
I heard at least the Income Proof is needed.
You are right. You can take over 5,000, but there is no limitation on how much you are take over it without cause troubles. So it is suggested not to take too much.