I'm not sure that the age is even the main issue: if one parent does not want to accept parental responsibility, how can the child be left to stay overnight with them, no matter what their age?
6 hrs/2 weeks isn't a lot of time, as others have said, but then, if you're not wanting to accept parental responsibility, then it's technically like leaving the child with a friend.
Maybe if a parent's name is not on the birth certificate, there is no way to get that adjusted - I don't know the system - but if you do something as huge as rejecting your child at birth and then, only 4 months later decide that you're going to be a good parent, it shouldn't be as simple as clicking your fingers.
The other parent's questions should have been more along the lines of, "I would like to be more involved in the child's life, how can we go about this that would work best for the child". It sounds like the intention was to satisfy a desire (whim?) of this parent, not focusing on the best interests of the child.
Incidentally, if this parent is serious about being involved in the child's life, it's whole life, not just when s/he feels like it, why not suggest getting a formal arrangement set up - at their expense. Presumably that would focus this person's mind a little more on the seriousness and responsibility required of being a parent. It would also put you in a more certain position if the other parent starts missing appointments or extending visits etc in the future.
Good luck!