I agree with bloodread4, that is correct about Kid's mother but my guess was this: Shingami-sama fell in love with a human but when their child, Kid, was born Medusa attacked Kid but she had to flee because his mother and Shingami-sama rushed to his room because they heard his crys so she desided to come back when Shingami-sama was not nearby. Two moths passed when Kid's mother had taken the two month old Kid to the store because Shingami-sama was still at the school but he would return soon so she was getting some stuff for dinner so then Medusa attacked them. Desspret to save her son's life Kid's mother made a deal with Medusa. Medusa could take her life in extange for Kid to live. Medusa agreed. Now to take the woman's soul she would have to realise her soul protect thus alerting Shingami-sama of her presence and the fact that she was attacking his family but she did anyways. Shingami-sama sensed this and rushed to the back ally that the attack had taken place in but it was too late, his beloved wife was gone. He was so upset he didn't sense his son but when he heard Kid's crys he was overjoyed. He took Kid home and raised him on his own but he never told Kid what happend to his mother and that is why he tried to hide some of the truth from Kid was so he would be able to protect him and so he wouldn't lose his baby boy. Thats just my guess though not what actually happend.
Death the Kidd is a shinigami. Shinigami's have fragments attached to there soul when the fragment becomes strong enough it detaches. Both Asura and Kidd are from the same fragment making them brothers. Because Asura is the Keshin of madness Death The Kidd's madness manifested into OCD. When Kidd gets old enough he will eventually have to surpass his father and become a full Shinigami and take his place. The cycle from there will keep going
Death the Kid didn't have a mom. He was created by Lord Death because he was working on a soul experiment he then was born. Thats why he's OCD (or such a perfectionist). Since he didn't have a perfect family then he would at least make everything else perfect.