SYNCBACK
This is by far the best I have tried. And it is free. There is a Pro version available but I can't imagine what extra features you would need above the free version. This will do all that you want, although you may have to create a simple command line to run it on startup. There are some helpful forums for this application and some people have posted batch files that you run on shutdown. These batch files will run the backup/synchronisation and then shutdown the PC. This seems a far better point at which to do the backup.
Hope this helps.