Okay. I got the impression from you saying how you would like to respond and your opinion of them.
However, you can find out if the letter is from a genuine solicitor by asking either the SRA or the Law. Society (I can't recall which) if they are registered with them. If not, then it's a fake. I did this in past when someone pretended to be a solicitor with me
How close is the trampoline to their fence? I've known people who've put them right beside the fence and those on the other side have felt really threatened by them and found them very intrusive. I've seen those issues dealt with by the trampoline being moved away from the fence and if they are as unpleasant as you say, then that would probably be the best position. On grass, you can also anchor the thing.
You're probably going to be living next to them for a while and it sounds like they can make life difficult for you, so I'd be looking to try and not give them an excuse to do so.
That said, I'm a bit of a stroppy bugger and if I received that letter, I'd have been round to see them (after checking if it was legit). I'd also have done something after seeing them videoing because I am fairly sure that is an invasion of privacy, assuming your daughter was in your garden, which I think you said was the case.
By the way, citizens advice will help with things like this. They can report you to your local authority and you'll know best how likely your authority is going to be too do anything, but if they're making up solicitors (and spelling mistakes from solicitors are not as uncommon as you'd like to think they are (married to an ex legal)), then they don't seem to have a clue.
I would also say - easier said than done - try and step back and do something else and sleep on this. You will be able to make better decisions when less emotional and that will only help you.
Hope that helps.