Anyone who is born in the United States is eligible for presidency, it does not matter the birth place of your parents, only you.
You don't have to be born in the United States to become president but one of your parents must be a US citizen. For example, a child of a US citizen born on a military base in Europe can grow up to become president. See related question.
The Child is automatically given American Citizenship and if the parents are from out of the US or Migrants they are given Legal guardian status and can stay to care for and provide for the child. Also the Child is entitled to the health care offered to mothers and newborns as well the ability to bring others into the nation to get citizenship by age 18. Also the Child is allowed to become President or Vice President.
First of all this is a stupid question but they become parents because they want a child.
Zeus became the important child of those parents because he was gay.
when the two parents that are dating become married and are legal
Some countries will claim the child has dual citizenship (see links below), however, while the US does recognize dual citizenship, any child born to US parents ANYWHERE is considered solely a US citizen
By allowing the parents to vote for the child. Parents act as the representatives of the child in all legal matters. If for example a child owns shares in a company the parents decide how the child votes on issues concering the company. Voting for president could be arranged the same way. In the U.S. this is currently not allowed.
unless the law is changed the answer is yes if the child was born in the US then he/she can be president regardless where the parents are born.
when they are named after one of their parents.
It depends. But it is likely - if the parents petition on behalf of the adult-child.
100% yes if born in an overseas US military hospital. If born in an off base establishment, the local base/fort/camp will have paperwork for the child's US Citizenship. In some countries, for example the UK, when the child is born off base they can be granted dual citizenship, as my son was.
In most civilized countries that is not legal, therefore: No.