okay, thanks. So as far as you know, the shield should not be piggybacking onto the laptop's network connection...via the Arduino USB.
I think the wall wart 9 volt works, but it was not connected to a UPS....and brown-outs are absolutely possible around here. (lights dim momentarily when AC kicks on).
On #1, no. The w5100 has no way to connect to the Arduino. The Arduino is the SPI master and the shield is the slave. Unless the Arduino contacts the w5100, the w5100 has no way to communicate with the Arduino, except an interrupt on pin 2 that the library doesn't monitor.
On #2, that is not related strictly to Arduinos. My Mikrotik routers suffer from the same malady down to the ethernet port failing when the power browns out. I must reset them to get them started again. Needless to say, they are all on UPS devices now.