Something jumped out at me so I didn't read all the replies, but here's a thought. It seems that dss wants to spend time, but not overnight. That would scream to me, "SOMETHING BUGS ME AT NIGHT!!" Have you asked him specifically what about the nights bothers him? Maybe babies crying keep him up, maybe his bed is more comfortable at mom's house, maybe you all shut the house down before he's ready to sleep. I don't know, but I would want to find out what specifically about overnights bother him. It could be something as simple as the house shutting down before he's ready, or it could be something bad like he's being guilted by mom about him being the man in her house and she fears sleeping without him there. But until that question is answered, I think you're going to have issues.
I would also stop stressing "make up" time and simply stress that you all want to spend time with him. Alot of this sounds like typical teenage behavior, back talking, snide remarks, testing boundaries, all sound very typical of a teenager. I know dss is going through alot of that right now. Learning how to talk to a teenager and really address the issues in a non-confrontational way that allows them to open up is VERY important. DSS and I have had alot of practice, but even so, we still have times when we both have to walk away and try again later.
I know it isn't possible for me to separate from DSS and let issues be between "dss, his father, and his mother". It doesn't work that way for us. Everything that happens with dss effects me and the three other children in this home. So I've never been one to separate, but I do love this one. If I don't take an active role then I get accused of not caring enough. If I take an active role then I get told it's none of my business. I feel caught in a catch 22 most of the time, so I just don't listen to those comments. DSS is a part of my family and I care what happens, and I refuse to aplogize for that. I have been his stepmom for 12 years and I love him dearly.
Hang in there, learn to talk to him, reallky talk to him and try to see things from his point of view. Avoid issues that create a conflict of his loyalties (such as making an issue out of his staying with mom over night, you never know what kind of pressure he's recieving from that end). Become his ally in this, rather than another antagonizing factor he has to deal with. Set boundaries but make it clear that he can always discuss the boundaries if he feels they are unfair. Find out why he chooses one thing over another. Once you find his motivations you can sometimes find a compromise that works for everyone.