Lydia - you say 'he can do what he wants but I'm not allowed to'. But this isn't true - you are 'allowed' to - he is just making life difficult for you. So you need to tackle this step by step.
As everyone says get proper legal advice and get a proper custody arrangement set up.
You are of course right to let your son see his dad as often as he likes - as long as you can control where he stays at night ie. he cant just decide to stay there without your permission.
If you make it clear to your son that you will always allow him to see his dad, would he be calmer about moving/ making sure he comes home when he has to? I think as someone else said, get your current partner on side and have a really calm chat with your son, about why you need clear structures in place.
The poor kid is under horrible emotional pressure from your ex - so try and ignore the teenage tantrums
It's great that you are supportive of your son seeing his father - you are clearly a good mother and have done well to get away from this odious man.
But while continuing to allow him to see his dad, you can get a clear idea of what you can do yourself as mother.
I think you also have to separate normal teenage behaviour out from this - I was hideous to my mother from about 13 to 18....! he will play up and you just have to not sweat the small stuff - its a tough situation for him, let him have his tantrums and just be very clear what your rules for him are.
You really have to stop giving in on disciplinary issues when your ex intervenes - thats bad for your son.
It sounds very tough - but the sooner you get legal backing for your guidelines then set them clearly the better. Make all contact with the ex very official.
Your son isnt going to grow up to be an abuser just because he shouts at you! that is really unhelpful advice - all teens are nightmares sometimes. He is just a kid still.
anyway, good luck, be strong and believe in yourself. and just accept there will be bumps along the way, but dont give up