Hey John.
So what we've found hasn't been straight forward by any means. I've been using the AutoCad 2017 trial for a few days now, and this newest version of AutoCad was signed into my account from the start and remains as such.
I went full-on OCD in this effort and asked everyone on our network if they were experiencing the same hangs that we in the engineering department were. As it turns out, they were. Now our CEO (who operates as primary in-house IT maintenance), myself (arguably secondary IT maintenance), and our contracted outside network service provider have all been working diligently to try to identify the process(es) that is/are causing the hanging.
This further digging revealed some company network gremlins living on the server leftover from a major hardware swap. I wasn't involved in that upgrade, so I cannot speak to the specifics of where or how these gremlins were present.
Of the list of sites you gave me to ping, I am still finding the sites I mentioned above as timing out.
All of that said, the hanging has decreased in both frequency and duration. Seeing as we're currently running different versions of AutoCad (2012, 2017LT subscription, and 2017 trial) and experiencing system hangs at the same time, I'm more inclined to think that the inability to sign into the software was because of the combination of AutoCad 2012 and Win10. As for the hanging, I think we found reasons other than AutoCad for that.
I appreciate all of your time, and everyone elses, but I feel pretty confident in calling this issue addressed. I might even go so far as to apologize for blaming the network issues on 2012 as, to my knowledge, we haven't found any processes related to Autodesk products that definitively caused the network issues.
Now if we could just get HP to service the brand new plotter they delivered to us in November!
Thanks everyone!
John, seeing as the only issue that we did find that made a noticeable difference (and likely the only one an Autodesk product had a problem with) was the time syncronization, I have accepted that reply as a solution. I hope that's okay with you.