This is my first post and so haven't read the entire thread but wanted to say something about the possible involvement of Social Services. I am a retired social worker/manager in Children's Services (a career spanning 30 years) and I note some posters have suggested involving them, to try to get your exH removed from the house. I don't think there is any way they could do this to be honest.
Firstly you would need evidence that he is harming the children (how old are they by way) and they would almost certainly suggest that you leave him with the children, so that they are safe. They might carry out an investigation (known as a S.47 investigation) where there is good reason to suspect that a child/ren is suffering significant harm or likely to suffer significant harm. IF they did this (and I say IF because social workers are absolutely inundated with work and because of all the budget cuts they are not even able to remove children from unsafe homes as they can't afford to look after them. Even IF they carried out an investigation I'm sure the ex could put on a good show of him being nothing other than a caring father. Unfortunately horsetowater things are not as simple as you might have been led to believe. There has to be hard and fast evidence that a child is being neglected or ill treated and leaving videos around or on pause is not going to provide such evidence.
Having said all that I am wondering what this Order is that you have in your possession (sorry it was probably explained upthread) Is it a "non molestation injunction" - if so does it have a "power of arrest" attached because if it doesn't, it isn't really worth very much to be honest, although you do say he could be removed from the house if he is affecting your emotional/mental health and this must surely be the case.
The trouble is there are no easy solutions even with injunctions because often these men will just return to the property and be 10 times worse because of having been removed, if you see what I mean.
I think your response to his e mail was very good and very rational but I don't think it is going to help, as you are trying to meet his irrationality with rationality and this never works. If he was able to be rational you wouldn't need to be communicating my e mail.
It seems you are in a Housing Assoc property - is that right? Are you joint tenants. I think you do need to consult a good family law solicitor without delay but the problem is as you are aware that there is no longer legal aid available for these private law matters (thanks to this govt) and getting the matter into the Family Courts would mean you either paying for this (costing a great deal) or representing yourself. Needless to say most people would shy away from that and this is the reason there are fewer private law cases going through the courts. It may be that the ex could afford to pay for legal representation and that would be the worst of all worlds.
I know he should move from the house but I don't think he's going to do this to be honest. Of course you should be able to stay there with the children but you are going to need help, because as you say your self esteem is very low, with small wonder as I'm sure this is all part of his plot to undermine you. He sounds potentially very dangerous.
I think after seeing a solicitor your best bet would be to contact Women's Aid (you can find the number if you google) as they are used to dealing with cases like yours on a daily basis. It may be that you and the children have to leave the property and go to a refuge, at least until the matters of property/finance have been sorted. I only say this because I don't think there is a cat in hell's chance that this bloke is going to leave. You are being emotionally abused and need to get away for your own sanity. Sorry I've just thought have you any family/friends with whom you could stay.
The other charities that are very good are CAB and SHELTER but I don't think you are going to be able to get anything sorted until you are away from this man, and leaving now doesn't preclude you returning and him leaving, but I think first and foremost you need some space from this man. IF you do take this route, please don't tell him - collect the necessary stuff and go (Women's Aid will help with this) I don't want to be alarmist but many men escalate their behaviour to causing severe harm (even death in some cases) when abused women say they are leaving.
Sorry if I am wide off the mark here but I am just trying to be realistic.