I heard that Sarah had to conform to a custom of married women, and wear a headcovering that covered up her eyes 20:16.
I don't think this is referring to a literal eye covering, but is figurative. However, it is very interesting to consider in light of 1 Corinthians 11, and I hadn't noticed it before.
Genesis 20:16 said:
And unto Sarah he said, Behold, I have given thy brother a thousand pieces of silver: behold, he is to thee a covering of the eyes, unto all that are with thee, and with all other: thus she was reproved.
I think Abimelech is saying that Abraham should be a figurative cover on Sarah's eyes, so she does not look at or go to another man. She should keep her eyes only looking at him, to keep her from looking at "all that are with thee", ie all other men in Abraham's tribe, and "with all other", ie external men such as Abimelech himself.
However this does also directly parallel 1 Corinthians 11, where the literal covering is a representation of the husband's headship.
I also heard that the covering up the eyes and face was a sign of prostitution - Tamar and her father in law.
Genesis 38:15 does say he thought she was a prostitute "because she covered her face", but I think that means "he thought she was a prostitute because of where she was sitting and what she was wearing - and because she had covered her face, he didn't see she was actually his daughter-in-law". There is no direct indication in the text that only prostitutes wore veils.
On the contrary, Genesis 24:65 shows Rebecca veiling herself, and she certainly wasn't a prostitute.