I'm afraid you cannot do it, or at least you cannot do it via the home sharing mechanism at least.
Certain items of Meta data, (including play counts, last played, ratings etc) are not stored as a part of the songs own Meta data (Title, Album, Track number etc), but instead belong to the iTunes library as a whole. When you transfer a song using home share, you are effectively making a copy of it and placing it into a second library, where the flags are effectively zeroed out as if they have never been set (which, they haven't, for the new library that is).
That's just the way iTunes works, it all revolves around your library as much as the individual items within that library, so a play count would not copy over as it is not considered a part of the song itself.
You may be able to script up a method of transferring such library Meta data, but it's by design that you see this behaviour using Home Share. Tools may exist, but I'll leave it to others that may have used them to mention them as I have no relevant experience.
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.
Best Answer
Per this thread from the Apple Developer Forums, it seems to be a source of much confusion for developers, however the default behaviour is described here, which states that the summary rating is specific to each territory and can be reset if the developer so wishes upon releasing a new version of the app