Firstly is it possible to use objective C to find out if the user’s account belongs to the US, UK, Australian… store?
Secondly is it possible to get the user’s iTunes account name in order to identify the user across multiple devices? Or is there a useful alternative that doesn’t require user registration?
No, and no. Sorry.
You can guess their store from the device locale. There may be a few Australian users who use the US store, etc, but probably not many.
To follow a user across multiple devices, saving a record in iCloud seems to be the best option, or using some other login-based system like Game Center, but both of those are opt-in services, so not all users will have them enabled.
Is this is for analytics purposes (i.e. to gather general stats about your user base)? If you use something like Flurry then this can track users across devices by cross-referencing the stats from multiple apps against the device UDID, but again it only works for a proportion of users, not all of them.
If you really need this info for the app to work, just ask the user to enter their email address and iTunes store locale when they first launch the app, or make iCloud usage a requirement for the app to function.