IOS – Privacy: can a app from App Store access the Apple ID data

Your mother's iPad will take you through the process of creating an Apple ID, but you can also create one on her behalf at Apple's website. Whatever you specify as the country of her mailing address there will determine the Apple Store she becomes associated with.

So the answer to your first question is yes, it is based on the Country/Region you specify when creating a new Apple ID.

The answer to your second question is no, it doesn't matter what you set as the regional localization of your iPad itself. You can change the iPad's localization as often as you like, indeed I do this when moving between the UK and the US. The iPad localization determines things like date and telephone number formatting conventions, spelling dictionaries, and so on.

As an addendum, you can have multiple Apple ID's on a single device. This is kind of annoying to do, but sometimes it's the only way to get an app. In my case, my main Apple ID is linked to the American store, but my London bicycling maps are only available on the British store. So I maintain two different Apple IDs, and when I update my apps I have to type in both passwords and juggle them a bit.

Your mother may not want the hassle of doing that, in which case you are well-advised by @stuffe to keep everything on a single ID.

There is no publicly available data on the actual iTunes App Store download counts (except for occasional special events such as the 25 billionth app download contest, and a few individual developer reports). The exact app download numbers are considered confidential information between Apple and each developer (perhaps because an individual's earnings could be so deduced). Each developer has access only to the counts for their own apps.

That said, a few developers do report some of their download numbers; and there are various estimations of the download distribution curve based on this evidence: for instance this blog post, and this academic paper, both from 2011, which may or may not be accurate. (Disclosure, the blog post is mine.)

As for part #4 of your question, the "on average" answer may not be relavant, as the app download distribution may be so skewed that almost no app has a download count that is anywhere close to the mean average, but mostly a lot more or a lot less. For instance, whatever the big mean average number is, there are some reports that most apps get downloaded less than once per day.