I agree with you in that Rollo has some psychological need for family, a desire that's going to get in the way of him and his mission. Personally, I don't think Rollo has his own birthday, either; he's too obsessed with the mere concept of it for him to have one. He's grasping on to any familial ties with Lelouch and as a result, that's compromising his judgement with carrying out his duties.

It's a photo of Lelouch inside. With lots of sparkles. COUNT ON IT.