1. Right click on your device in iTunes and click "transfer purchases".
2. For the visual badge on the Mail App icon: Under Settings>Mail, Contacts, Calendars set "Fetch New Data" to Push (may take more tweaking depending on your mail account) For the silent part go to Settings>Sounds>New Mail>Off
3. The stock mail app doesn't offer a week view, should be coming with iOS5. There are 3rd party apps that have this feature, have a look at GooCal.
4. This is controlled in iTunes and there are different ways to control what music is and isn't synced to your device, too many to cover in a short answer. Typically you would uncheck playlists, artists, albums, and songs you don't want to sync. We can cover more details on this later.
5. Sorry can't help
7. Feature not available on a stock device. Requires a jailbreak if there is even a tweak to do that.
8. Someone else will be better equipped to field this one, I only have one of these devices. Typically each device is handled separately in iTunes. iOS5 will be changing this somewhat.