Take a look at somewhere like Caswells hire scheme - think they do it by post. Particularly if you're getting a 1/4 for a 7 yr old, you may want to upgrade sooner rather than later. But if not you can offset hire cost against purchase price.
My 6 yr old is on a 1/2, she's a bit tall but would be average for 7 I'd think.
We've used cheap bows from eBay or Amazon, they're fine. We usually find even when we buy a small instrument with a bow the bow hair is worn out and they're cheaper to replace than rehair.
Also, does your young teacher have her own teacher supporting her in this? Teaching violin to beginners is hard. Assuming she does have a teacher on board, perhaps that teacher might have pupils moving up in size, or at least have some violins herself so she can try out to find the best size for you. Perhaps you could see if you could arrange a one off session with teacher's teacher.
Alternatively could you ask your local shop to try some for size? I don't think it would be cheeky if you're going to give them business setting up.
But I think once you've paid 30 for violin, plus setup, plus strings, plus bow, plus any extra repairs needed - something like Caswells with hire and purchase option on cheapest range stentor would look attractive financially.
Also look into cost of group lessons and instrument hire via school and local music service. You might be pleasantly surprised.
Sorry, rambling thoughts there...!