At some point, with stroppy neighbours you have to just decide what is reasonable and get on with it. It is reasonable to shut doors any old way you please in your own home, it is reasonable to have a musical instrument and practice on it, it is reasonable to have kids that cry or shout from time to time.
Our neighbours called at our door and asked if we could not do DIY on the weekend as this was their relaxation time! Um, no, it was the only time we could do it as we were at work all week. You have a perfect right to use power tools, hoover, have a dog that barks occasionally and so on.
Environmental Health will not be interested in any of these things it's only if very unreasonable (like dog who barks 100's of times a day, late night noise, very loud music).
Just get on with your lives, you can't let these people tell you to live a different life in your own home. They have a dog, they have a hoover, they also make noises, that's the way it is.
As others have said, just go about your business normally, you are allowed to close doors any which way you like in your own house including slamming the front door if you choose (I don't mean hard, I just mean shutting it firmly). Stop caring what they think, stop paying for stuff and tell them you are now writing down every time they contact you and are going to report them for harassment.
You have tried reasonable, these people are not reasonable, don't try any more!