First , i think there is an error here :
01:09.0 Ethernet controller : Realtek Semiconductor Co., Ltd. Device [10ec:8161] (rev 10)
PCI ID of your card is 10ec:8161
Full list is : PCI\\VEN_10EC&DEV_8161
10EC is realtek ID , with device model 8161
i check asus website and download driver for windows to see if PCI ID is correct ( your card is NX1101 V2 )
You can open rt64win7.inf , and you will see no PCI\\VEN_10EC&DEV_8161, and I think your card is realtek8169
If this is an error detection, you can try move your card to different PCI port in your Mainboard and check lspci -nn again.
and i think the linux driver in asus website not specific for fedora, because package format is tar.gz ( fedora package format is RPM )
You can examine physically your card to see what chipset in use.
for drivers : http://www.realtek.com.tw/downloads/dow ... Down=false