1. No, this doesn't get you off the LTS--you'll be running the LTS but with newer Cinnamon or MATE version. That's what backports is for
But that is a side-effect you might not want if you just want the newer minstinstall version. Upgrading Cinnamon or MATE on the LTS can potentially give issues (there have been a few here on the forums; but of course on the forums we only read from everybody for whom something went wrong and not from everybody else for whom it worked without problems).
2. If you download it from the website yourself, and try to install it, it might complain that it needs a newer version of some other package also. If you use the backports trick (clear all in Update Manager and only mark to upgrade mintinstall), the Update Manager will try and handle that for you automatically--by also marking any other packages that need to be upgraded. It's safe to first try the manual download though.
3. I think so yes, but I don't have Linux Mint 13 here any longer so forgot how you can find it in your menu. In Linux Mint 14 you just have 'Software Sources" as an application in your menu.