Can you connect to the robot and gather the log files through ethernet or FTP connection?

The quickest way is to use a USB stick with the log magic file on the root of the stick. The magic file can be found on the UR support site. 

It it is possible to SSH into the robot and then get whatever you want from the files. We've mounted a remote directory on the robot before, I didn't actually do it so not super clear on what it took to do it but it allowed us to access files from a different computer. 

Thank you matthewd92, would SSH be done through ethernet?  I'm currently doing the USB and magic file route but I'm looking for a way with less of a repetitive manual task

Yes, its done over the local area network, need to make sure your computer and robot are on the same network.  I think on windows I used Putty to connect to the robot using SSH.  This will give you access to the terminal on the robot, then you will just need to navigate to the files.