1. Site error occurs because of electromagnetic conductivity properties around the transmitter (that is to say; magnetic fields, buildings, terrain etc etc around the VOR itself). This will be more or less constant over time, but may of course vary significantly on various radials. Propagation error adresses the propagation qualities of the signal, ie atmospheric conditions such as refraction, bending, scalloping etc. We know that electromagnetic waves are supposed to travel in straight lines, but various conditions affect this.
2. I'll leave that to someone else. I'm not sure.
3. The VOR does not suffer from quadrantal error. In fact, the quadrantal error has to do with the way the ADF (the on-board equipment) senses the relative bearing to the station -- so because the VOR reciever only recieves the the radial information based on phase difference, it does not compute a relative bearing (although it can be displayed on an RMI), and does as such not suffer from quadrantal error.
4. This would be equipment specific, but normally yes. The failure flag would normally be displayed anytime valid signal information is not recieved, for any reason.