>>In Genesis 5:2 for instance it says "Male and female created he them; and blessed them, and called their name Adam, in the day when they were created.
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>Two questions. First, was Eve's name Adam, too? Second, what does it mean "call a name something" - a name has a name? Which brings the subquestion: what were their names' names before that?

That is what it says, they both were called Adam. This was the only name that both of them were called by their Creator. Most people don't know this because they don't know how to read. The name Eve was not given by God, it was given by Adam after they were thrown out of Eden, and after she bore a child. That was mentioned in Genesis 3:20 "And Adam called his wife's name Eve; because she was the mother of all living." So it also explains the meaning of "Eve", but indeed this was not the original name of the woman and it was not the name that God gave to her. The same concept is still used today, so when you fill out the "name" on a form it means the "Family name", not the first name, the first name is a separate field. In most families the last names of the husband and wife are the same (nowadays they are sometimes different), so it's actually very common practice.

I think to call somebody by a name is an old fashioned way to say "to give a name". In German this expression is still in use. So when you had a child you "call the name of the child Joseph", because the name was used to address someone. Nowadays you use Facebook, so when a child is born you "create a new Facebook account with the name Joseph". More advanced families now also create a Twitter account and the newborn is tweeting about its experiences and early traumas.

Because Adam and Eve were just born, if you will, they did not have a name before that. So the text refers to them as Man and Woman before that. They shared the same pool of genes because Eve was made from the genes of the bone marrow of Adam. Most people thought this was not scientific, but now they do know that it is scientifically possible to clone a person, and the best source for that would be the bone marrow. That's why they are now so fascinated about the finding of the Mammoth's bone marrow.