1) Well, it's more complex than just running one file. It's actually run from the Windows Update website, however, it does check some files on your computer for reference, etc.
2) Yes, it will repair / restore corrupt files to an extent, but it is not a cure-all for any system file(s) that may be a problem, etc.
3) The most common command used is "sfc /scannow" (without the quotes).
4) I think it would be a waste of time as I'm leaning to there being a nasty rootkit that was loaded into your OS somehow, and as such it would be much too advanced for you to try removing it even if you had good instructions, etc. There are many different rootkits out there and no one size fits all removal (if even possible) procedures. More often than not, the end result will be a complete reinstallation of the OS on a reformatted partition / hard drive.
There are even variants of rootkits that'll infect hardware such as the BIOS or boot loaders, in other words.....they are nasty!!
I've included a link below for the Wikipedia article on them if you care to browse through it.
5) My advice to you is to take it to a reputable repair shop, or 'try' reinstalling the OS first to see if that'll work.