grep RealtekRTL8111 /var/log/system.log
Apple reworked logging completely in Sierra. You need to use a different command in order to retrieve kernel logs. Please see post#1 of this thread for instructions.
Anything I should concern myself with your network drive on newer versions of OSX?
Honestly, I have no personal experience with the driver on pre-SB hardware but in theory it should work fine. The only thing I'm concerned of is the fact that some users with old hardware reported performance issues due to power management which was reworked by Apple in El Capitan and Sierra.