From her - exact quote: "The written agreement that we did before I left DID, in fact, spell out going to TN. We BOTH signed it and, though it was not notified or filed in court, the attorney from Neighborhood Legal said it is a perfectly legal document."
That certainly changes things around significantly. Although the failure to get court approval
could become an issue, if there's an agreement between the parties saying something to the effect of "Father is aware of Mother's proposed relocation to TN, is fully informed regarding it, and through this agreement consents to both the the relocation and any changes in visitation/parenting time," then the court approval thing isn't as big a deal (if it's an issue at all - it may not be).
By "notified," I'm guessing she meant "notarized." That, of itself, isn't a big deal either, unless it's required by PA law. Notarization is, at the bottom line, verification of signatures by an independent party. Although it's a good idea to do it, simply so a party to an agreement can't say "that's not my signature," there are other ways to prove a signature is valid.
The unfortunate reality of family law: children can become weapons that one parent can use against the other. It is a horrible shame, one that I feel should carry criminal violations (how you prove it, I don't know); but, trust me, it's obvious to a judge when it's seen. It's one of those things that's not easily defined, but that's known when it's seen.
Keep in mind that, from a court's perspective, these things have to be taken seriously. If one parent says in a filing, "the other parent is an alcoholic, and she drives the child to the bar with her, and then drives around hammered with the child," a judge has to take that allegation very seriously when it's made - it's why these things are set for expedited hearings because, if true, that's a big time problem. And, if it's not true, that also needs be resolved as quickly as possible (with appropriate sanctions against the parent acting in bad faith).
Unfortunately, because
mere allegations regarding a child's well-being
must be taken seriously (if you don't, you're setting yourself up for a boy crying wolf situation) at first, it's a system that's begging for abuse by those who would do so. And it is abused. But, when it's abused, courts are pretty good at sniffing it out (trust me, it's obvious), and also pretty good at coming down hard on it. Here, if the father's acting in bad faith, he's going to end up paying for it one way or another.
[edit:] Again, I continue my disclaimer - not licensed in PA. But, these are general principles of family law relating to children that, to my knowledge, are applicable nearly uniformly across the nation.