This is what you should be doing not him!
Please stop worrying about the threats he is making - this is BS and just to intimidate you into letting him have things his way.
You need to stop giving him an audience. He loses "control" over you if you're not taking any notice of his crap. Use an email address for correspondence about the DC only. If you can, get a cheap, pay as you go phone and use that for emergency contact only. If he contacts you and it's not an emergency, tell him to put it in an email and hang up. Don't answer if he calls back - he will probably try to push the boundaries here and you have to push back. Block him on all other phones/media. Don't let him view your media profiles etc. Don't answer any personal questions about you (how you are, what you are doing, etc), you are no longer his business - just tell him that if he asks or just ignore the question.
You need time away from him to start to heal. He is deliberately being in your life/face as much as possible to try to prevent you from moving on and getting past this as he wants to keep you as backup. Make sure when he has the DC, you are out with friends or doing something special just for you.
CMS claim asap to make it formal and stop him using money as a weapon.
Tell him he can have DC EOW and one night (or whatever is sensible for the ages of the DC) and that he must stick to it (unless there is an emergency) as it is upsetting for the DC to be messed around. Every time he messes them around, send a formal email acknowledging what he has done. Don't any anything about how it makes you feel, just stick to the facts and how the DC have reacted.
Other PPs have suggested having someone (good friend/sibling) check the email messages before you view them to warn you if there is any nasty content so it doesn't come as a shock. It would also be good if they could answer the door as much as possible for drop offs and pick ups so he can't speak to you directly.
Remember he has cheated and lied to you. He has then tried to blame you. Only he is responsible for his actions; you aren't. Don't trust what he says as he is capable of lying.