My heart goes out to you, Summermary! IMO, you need to some issues in your mind, if only to bring yourself some peace. First, though I know it's frustrating that the dad came back into the picture a little late, IMO, it's good for GS to know his dad wants to be w/ him. Second, while I know it hurts to think that GS might be influenced by gifts and skating outings, etc. w/ his dad, IMO, it's really no harm if they enjoys these things together. I'd be glad that someone can provide these things, especially if my DD couldn't and neither could I. Also, I would be pleased that his dad can now offer a more stable environment than before.
What would concern me would be the dad's refusal to accept the ADHD and Asperger's diagnoses. Also, that DD seems to be so disorganized and confused about her communication w/ her ex. Clearly, she did not send him a choice of dates, even if she thought she did, even if only b/c of her phone. I know it's hard b/c of he ME, but she needs to get on top of this or let a solicitor or social worker handle it for her. Or Home Start, which was a good suggestion, IMO. The fact that she has trouble dealing w/ court, etc. is just all the more reason, I feel, that she can't handle this alone. And neither can you (nor should you have to). It's so beautiful for you to be there for her, but she may need additional and, perhaps, more professional help.
I'm also concerned about GS being in the middle of all this conflict, of course. It seems as if "one minute" he doesn't want to see his dad more often and the next, he's excited about skating w/ his dad, etc. I'm getting the impression that he's confused (understandably) and, maybe, sometimes, says what he thinks the parent he's talking to wants to hear. (So - well, you would know better than I, but maybe the dad didn't "lie" about things GS said, perhaps the poor child is telling each parent a different story.) GS needs to know that he can voice whatever is truly in his heart, no matter what anyone else thinks/wants.
In addition, I share your concern about the dad calling your house, "Nana's house." Is that just how dad happens to think of it or is he deliberately trying to change GS' view of it? But as much as GS is w/ you and DD, I'm sure he will continue to see it as his home, as you told him. Words are one thing, but his dad can't change how GS feels about his home inside.
Finally, I'm worried about you. You have - and have had - so much on your plate and you tell us you feel as if you're on the verge of a breakdown. This is another reason to get professional people involved. IMO, you've sort of become be DD's lawyer and therapist and GS' acting parent all rolled into one. And that's just too much. You need to take care of you. Hugs!