I do agree that the neighbor could have responded to your attempted communications, if for no other reason than to say "i'm looking into myself, so you don't need to handle it." or something like that. I also think it was a bit rude of to randomly show up expecting you to pay on the spot...not a lot of people have $450 in cash just lying around to hand out. On her part, she could have handled it better, yes. She should have called/written/come over when she knew what the price would be, and given you a date she would like to paid by.
However, i'm not seeing how showing you estimates would actually change anything. She could have handed you an estimate from every company in the state, and the price would still be the same because she would have still picked the same company. How do you know she didn't shop around first? If it took six weeks for her to bring you a bill, I doubt she picked the first place she called. Maybe i'm just misreading or over-thinking, but I get the impression you wanted her to show you estimates so that YOU could pick which one she would use. If that's the case, that's not right. You don't know why she picked the company she did, and she very well may have tried to get a good deal. Maybe she had to use them because they installed her original windows, or she prefers them because a family member works there. Whatever the reason she picked them, it was her choice to make no matter how many estimates either of you got, or who she showed them to.
We had a BB shot through our bedroom about 2 months ago. Luckily, we live in an apartment and they fixed it for us, no charge. However, when we went to the office to report it, they were very upset and asked several times if we saw the person, what they looked like, etc. They wanted to know who did it, i'm assuming to bill them for the repair. And no, BB's don't generally cause major damage..no shattering, totally busted out window. But the hole was big enough that our AC was getting out, and bugs were getting in. Not something that can just be ignored, really.
I'm all for "kids being kids", and "accidents happen". I don't see anything wrong with BB/air guns in and of themselves. It's the way they are used that is the problem. If you are living that close to people, they don't need to be using them in that area. One window I would consider an accident without a second thought, but three? That seems less likely to not have been intentional.
OP, I think a lot of the negativity of other posters is coming from the title of the thread. Saying that you hate having neighbors implies the neighbor has done something wrong, which she hasn't. It may also be coming from the perceived attitude that it should have been up to you. You told this woman you would PAY FOR the windows, and you did. You didn't tell her you were going to pick out everything and do it for her, so she didn't agree to that. She agreed to let you pay. Count yourself lucky that she agreed to let you pay without taking you to court or something, considering what minor things people will sue over these days. Also count yourself lucky it was just some windows and not a person.