To me, the notion of 'punishing' a child who is under the age of criminal responsibility seems to belong more to Dickensian times than the 21st century.
Of course there should be consquences when children misbehave, but your little girl is still engaging in magical thinking and does not have an adult's concept of time.
Your dc is the middle child of 3 which is well-documented as not always being an easy position to occupy in the sibling hierarchy, and you have commented on the fact that, unlike her sisters, she needs 'firm treatment'. I'm well aware that parenting is a delicate balance, but I hope that your dd won't grow up to believe that she was treated differently to her siblings.
You have described her as 'not brilliant socially', and it therefore seems curious that you have chosen to cancel her Brownies and violin lessons as learning a musical instrument can improve a child's co-ordination and confidence, and the emphasis placed on team spirit in groups such as Brownies can only serve to develop her social skills.
You've also chosen to cancel these activities for what will seem to her like a very long time - certainly long enough for a bright little spark to convince herself that she doesn't care for Brownies and she's not interested in the violin, and I hope she doesn't find it difficult to re-engage once she resumes these pursuits.
The fact that there has been some other incident(s) involving telling untruths/tales to get other children into trouble may be an indication that there are underlying issues, possibly based around her need for (additional?) attention, that you should try to explore with her.
Do you spend much one-to-one time with her? Have you asked her why she wanted an 'eyeball'? What did she expect to gain from having this toy? Did she believe that it would make her feel special, or that if she showed it to other children they would admire her, or did she intend to give it to a child that she hopes may become her special friend?
If she had been able to buy the toy without your knowledge, how was she going to it explain to you? Or was she planning to hide it somewhere in/outside your home?
What compelled her to take money from her teacher's desk? Did she think about it before the committed the act (planned), or did it suddenly seem like a good idea (impulse)? As she was taking the money, did she feel that it was a wrong thing to do and was she scared that the teacher might suddenly come into the room and see what she was doing?
These might read as rapid-fire questions but I'm not proposing that you hold an inquisition, and the above gives opportunity to discuss how she would feel if it was her money that had been taken and to explore the wider moral concepts of right and wrong including not compounding a felony by failing to admit guilt.
Telling a child that you are disappointed in them can make some feel that that they can never regain your approval/trust, and a downward spiral can occur. I prefer to say that any misbehaviour isn't worthy of them, and that I know they can do better rather than have a child feel that they're not 'good enough' in any way.
For the record, if this were my 7 year old, the letter of apology would be de rigeur as would confiscation of pocket money to buy a suitable gift for the teacher. If a special outing/party/treat was on the immediate horizon they may not get to go unless I believed that they were genuinely sorry and contrite. Other than that, I'd give them lots of cuddles and resolve to introduce assorted moral topics/dilemmas in my general conversations with them.
I would take great pains to keep this matter under wraps - i.e. not talking about in front of other dcs, or to other family members/friends unless I could be absolutely certain that there would be no breach of confidence.
I'd also be looking at myself/recent family events etc to see whether something seemingly unrelated had led to my child stealing, as some children steal to compensate themselves for a real or perceived lack.