hello. My name is***** can help you with your question.
One way you can backup a Samsung Galaxy S3 is with the Samsung Kies software which is downloadable from their site. I'm not sure though if that will provide you what you need when you are done migrating. What password ***** are you being asked for.? That asking for a password ***** addition to what you have set up? Also, are you certain that you need to back up that data and that it will not resynchronize have to set up the phone with your Google account? Most the data does but I cannot tell you whether that one does or not because I do not know.
Rooting this device is very easy, so that may be the path of least resistance.
You can root it by either flashing TWRP Recovery in ADB from your computer, then going into recovery - when you exit, it will install root. Or, you can use chainfire's autoroot. Just search for XDA chainfire autoroot (android devs are funny about other devs linking directly to their work, which is why I'm asking that you search). All the instructions are with that in the XDA forums. I think this is the way to go - you're only talking about 15 minutes to get it done.