How do in-app purchases work for child accounts with Family Sharing?
Although they're related, you've actually asked a number of questions there. I'll deal with each of these briefly:

Question 1: This question is specifically off-topic. We don't answer questions like Why does Apple... or Should Apple... etc.

Question 2: Yes, it is possible you could get a refund. This will usually be in the form of a credit on your account.

Question 3: You need to contact Apple, not the App publisher.

Question 4: I suspect you have a few options. Since you have to contact Apple, I would just ask them what the best option would be. However, one option is to change the account details on the iPad temporarily so that it's your account that's logged into the App Store. Then try to make the purchase using your account. (You may need to delete the app and redownload it using your account). Once you've got the coins, log out of the App Store with your account and log back in with your daughter's. (NOTE: If you had to delete and reinstall the app for it to be associated with your account, this does mean you'd have to log back into your account again if/when she makes future purchases).