1. You want # of email badge on GMAIL icon? or you want # of GMAIL icon in the notification area? Both can be done ... for # of email badge look into using Nova Launcher. In the settings area you would see an option to enable it (cost some $$$ for the pro version for this setting). For # of GMAIL in notification bar ... only ROM I have seen do it is unofficial AOKP build for SGS3. you can try that as well. I am not sure about the stock samsung touch wiz launcher. I rooted and start flashing custom ROMs within a couple of hours of purchasing this phone
2. One thing you have to understand is that battery life is not a one size fit all for android. It heavily depends on your usage and what settings or background processes you have enabled. I know coming from iPhone you want to put all kind of widgets on the home screen and enable all functions like BT, Wifi and GPS etc like iPhone but Android is different. It allows you to have fine grain control over software as well as hardware settings. You should be getting decent battery life (i.e. moderate to heavy use should give you around 16 hrs with 2.5 hrs of screen on time) if not then try to remove some widgets etc. iPhone has built in power management for all these hardwares and you cannot mess with it. But in Android its open to you to change it to your preferences.
3. what kind of music files you have? mp3? FLAC? its possible that music files are corrupted when you try to copy over files from iPhone to SGS3. Take another look and verify that all files are intact and not corrupt. There is nothing wrong with the Sammy music player ... you can also try google play music app from Google Play. But this shouldn't be a problem of music player. Can you try to download another fresh music file from market or another store to see if that works?
4. Bluetooth shouldn't drop connection unless you move out of its range. In close proximity it should stay connected. If it doesn't then you have a defective phone. My first handset had some problems with Bluetooth because every time I tried playing music or pdocast over bluetooth it made cracking noise. I exchanged the handset and now there is no problem.
Hope this helps.