I have been the primary disciplinarian but my husband would step in when he saw that i was getting to a point of frustration. I am an elementary school teacher, so I feel that I am harder on our son. When I see behaviors that remind me of "bad behaviors" that I've seen in my classroom, I jump right on it in an effort to nip it in the bud. My husband is more willing to let things slide.
Your son may be acting out in frustration as a reaction to the absence of his father. Since the behavior only seems to happen at home, your son is not pleased with things at home. His laughing at you when you are trying to discipline him is simply a cover for the pain that he feels inside especially if he is/was really close to his dad.
Try this and see what the outcome would be. Tell him to draw a picture of his family doing something really fun that makes him happy in his own kid like drawing way. Then have him explain his picture. If he has left a parent out of the picture, ask him why didn't he draw that parent. Your goal is to get him to open up to you and discuss what is on his mind. If he draws himself and both parents, that's great. Display the picture so that you and he see it daily.
But as for his disrespectful behavior, that must be curtailed. Try focusing more on the positive behavior and REALLY praise him when he is behaving more appropriately. When you discipline him, the main goal of a disciplinarian is to teach rather than to punish. When he speaks to you in a disrespectful manner, stop him. Tell him the way he should respond to you and then have him repeat it. You will have to do this continuously every time he misbehaves until he realizes that he must change his behavior.
When he will not follow directions, then give him a choice to either do as he is told or continue with the time out: maybe sitting in a chair facing the wall for 5-10 minutes. It will seem like torture to him but it is more effective than spanking.
You must be consistent with the suggestions that I have provided. As an elementary school teacher, you are familiar with a behavior chart. You may need to utilize one for your son which focuses on the good behavior.
Hope I have provided some insight. If not, please ask further questions. Have a great Friday.