I had similar questions this week since I just got a new 830 as a side grade to my 920. See my post and comments here:
New Lumia 830 on AT&T - How to activate and setup? Restore from Backup?
1. I think you can go to Settings - Sync My Settings and select Theme and App Settings to On. Not sure if these need to be turned on for both devices but I'm guessing they do.
2. Not sure but I also want to know about this.
3. It should if you have Sync My Settings turned on for App Settings. Can someone confirm this?
4. I am also seeing this issue but could not find a way on the Windows Phone website to change the name. I see Lumia 920 as my phone but right under that name shows Lumia 830. Then I see Lumia 920 again as an additional phone. I guess this is because of restoring using the 920 backup. Maybe I should have just started with a fresh install.