Just reading this thread now -- great news that it sounds like you're dealing with it in a way that works for both of you.

Just a few thoughts to add --
Perhaps you would benefit from reading some of Dr Maria Montessori's writings. Much of it is directed more towards infants and preschoolers, but the precepts are quite universal. It was in this reading that I learned to BACK OFF and not over-help my young daughter.
You're so right that it can be just automatic to step in and help. But this can be coming from a place of over-protectiveness... or from needing to BE A HELPER... rather than actually providing HELP THAT IS NEEDED, if you see the difference.
I was an excruciatingly independent kid. I hated it -- and STILL hate it -- if someone thinks I need help before I've asked for it (and I hated asking for it too). It feels like a personal insult, as others have said. So it's a bit surprising that I had to be TOLD this via Montessori's books lol...
Anyway. From my daughter's early toddlerhood, we backed off and just let her make her own mistakes. If she got frustrated, we'd still just let her deal with it -- if we intervened at all, it would only be in affirmation -- "that's very frustrating, isn't it." Then when she'd ASK for help, we'd help.
Doing this deliberately was very enlightening -- DH had even stronger instincts to jump in and help his precious little girl. Avoiding doing it on purpose, we really, really saw how CONSTANT our desire to intervene was, and really realized that probably wasn't the best idea! How would she ever learn to do anything, to figure things out, if we're always doing everything for her and explaining everything to her before she's even had a chance to think about it herself?
She's now 3.5 and shows every sign of being as maddeningly independent as I was (am). "I do it!!! I do it!!!" But we've learned to celebrate that instead of fight it. She is VERY capable, very confident, she works hard when she wants to do something and doesn't get frustrated easily, she thinks about things and works things out. We can honestly say that learning to back off in terms of "helping" when it's not needed is the #1 most important parenting lesson we've learned with her.
Now, when it comes to homeschooling a child like that... I would say just use it to your advantage rather than fighting it. Find curricula that are more child-led, more independent studies. Or at least, things that you would be doing WITH her, rather than teaching AT her, if you see what I mean. You may even be able to back off with curricula altogether, or to a large extent, if she's that driven. She's only 7, there's no need to do a LOT of academics at this age anyway.
For the reversed letters, and mistakes of that nature, as has been mentioned, you kind of have two other options. One is to just let it go. She's practicing. Practicing means it's not perfect yet. If she were able to do all her letters perfectly every time, she wouldn't have to practice them. As long as she's not getting WORSE, I wouldn't even fret about it, it's a normal part of the progression.
The second option is to make her aware of the mistakes, but in a more gentle and fun way. Let her know there are 3 errors on the page and challenge her to find them herself. I've found that this works MUCH better with my son, for instance -- who isn't the hyper-independent type like my daughter, but who is VERY sensitive to criticism. Or I don't even tell him there are mistakes -- I just say, okay, now look it over and see if you can find anything that needs fixing. If I tell him "fix this letter" I'm likely to get screaming and crying and shouts of "but it's already perfect!!!" If I say "see if you can find anything that needs fixing" he'll smile and say "whoops, my 's' is too big!"
As for the piano lessons, I will speak as a piano teacher.

Quote:
| If I never offered her assistance unless she asked for it I can't imagine how that that would work. If she's practicing one phrase incorrectly for a whole week before her teacher catches it, that's a whole lot of time she's going to have to spend unlearning and then re-learning it. |
Yup, it is, and that's part of the learning process! If you never offered assistance, it would work just the same as it does for the vast majority of piano students whose parents don't have the first CLUE about how to help their kids musically anyway! They do just fine.
In fact, if you're helping her too much with her practicing at home, it could give the teacher a false idea of where your daughter actually needs help and where she doesn't. If you've 'fixed' all her mistakes, the teacher might erroneously think your DD did it all herself, and think she's ready for something more advanced when really she isn't yet, or there could be games and exercises your DD would benefit from that the teacher won't bother with because she doesn't need them.
Teachers want to know where kids are struggling, because they have the knowledge of the best way to fix it in the long run. When a kid comes in with a particular kind of mistake, yes it takes "un-doing", but it REALLY helps me to see how their learning is progressing and what's going on in their minds. Seeing a consistent pattern of particular types of mistakes really lets me know what to focus on in their development.
Sometimes I would assign something specifically to see how a student managed with it -- what they were able to figure out and what they weren't. I need to see what the student's level of comprehension is -- not their mom's. A teacher doesn't expect something to be perfect after a week of practice... like the handwriting, if it was perfect already, there'd be no need for a teacher in the first place. You're paying a piano teacher, let them do what you're paying them for.

If the teacher asks you to help at home in certain ways, then do that. Otherwise, let her make her own mistakes, because making mistakes AND YES, learning to FIX learned-in mistakes, is an important part of the process!
This has, again, been especially hard for me to do with my son, who is working on piano now at age 12. I give him lessons once in awhile where we do work on corrections, but when he's practicing, I leave him alone. It hurts my gut to hear him play wrong notes! But then something amazing happens -- he fixes them on his own! Sometimes not right away, sometimes it's days later. But 95% of the time he eventually figures it out himself. If he doesn't, then the most I try to do is give him hints. "You might want to have a close look at the third measure... just saying..." lol...
All the best!