Living with boyfriend

If you're living with your boyfriend is it normal that he asks you to leave so that you could have some space once in a while? Just for a night here and there to stay by a friend or family. I guess I can understand everyone is there space, but I'm always so reluctant to leave because he has cheated on me before and lied to me numerous times. So I get bad anxiety if I'm not there at night. We constantly thinking if you met up with someone or if he is talking on the phone to another woman. This is because he has done this stuff before when I wasn't there.  If I honestly knew he would just be sitting home watching TV I would have no problem with it, but it is literally so hard for me to leave because of my fears. And then he gets mad and says I treat him like his mother. But if you wants to gain my trust this is what he has to put up with. He says we spend all of our time together so it would be nice to have time apart which I know every couple needs. But when I do leave my head constantly rins and I want to keep cling hun to see what he's doing. I even have him send me pics of him sitting in bed to prove that's where he is. i know it's not normal and I really wish I could feel less insecure and less attached  cause he gets annoyed and says I'm always up his ass. And since I rarely do leave, a lot of the time we do spend together doesn't really seem wanted by him cause he very frequently hints that he wants me to leave sometimes and give him space. Other than telling me to leave him, what advice can you ladies give me to trust him better and maybe get him to seem like he actually wants to spend time with me? We are happy together most of the time that we are with each other. But sometimes he gets in these moods where all of a sudden he just wants me to be gone right then and there. And that is part of why I think maybe he made plans with someone. But it's probably me overthinking. His family even says that he is the type to want his space. I'm not really like that. I don't mind not being with friends or anybody else to be with him. That way I know what he's doing. I know it sounds like I should leave him but we do have a great relationship. This is just a bump that we need to iron out. So any advice would be greatly appreciated. Thanks! Sorry this is so long!