|The router shouldn't be. If your DC is behind it (ie: downstream) as it should be, then you would forward DNS on the DC and you could either:|
A) forward to your router's IP (which in turn forwards to your ISP)
B) forward to your ISP's DNS addresses directly (my preferred method in a case like this)
Not being able to bring up domain users on the share you wish to set access on usually means a problem with the AD and DNS with regard to the problem server.
If it were me, I'd unjoin the server (the right way) from your AD and then I'd join it again and see if that doesn't fix the issue.
You may also want to check the DNS host records and make sure they point at the correct IP for this server. If the host record for the server doesn't have the correct IP, that will cause a problem too.
You do have a statically assigned IP on the server..........right?