So your router and pc are gig - but what about the nas? What is the model number on your nas?
Just because your pc or nas has gig interfaces does not mean they are connected at gig - do the lights on the routers lan ports show gig speeds or 100?
5 to 12MBps would be normal for a 100mbit connection.
And just because your a fan of your AV software does not mean it does not slow down transfers. If your scanning the file(s) as you transfer them, there is going to be a performance hit no matter what AV it is. How big of hit is the question..
Normally gig speeds should be over the 30MBps for sure.. Upwards of 90, so here is quick drag and drop of a file to my File server
So while speeds your seeing are normal for 100mbit connection, they are no where close to what you should be seeing on a gig network. No jumbo frames on my network either, they are more trouble then they are worth in a home setup.
Try a robocopy of a large file - this will give you an exact number and time to work with. I don't always believe the dialogs reported speeds
Especially if your doing a large number of files. So pick 1 large file as your test.