After spending 15 to 20 hours on this over the weekend, I'm stunned to report that in my case the problem was resolved by switching from CAT 5 to CAT 5e cables, a solution I dismissed as not worth trying when I first came across this suggestion from another 5 hours into my efforts. Here are my setup and symptoms. If anyone's interested in a particular log or report on my setup I'm willing to pass this along.

Bought a new Shuttle DS61 v. 1.1 with dual onboard RTL8111/8168/8411 PCI Express Gigabit Ethernet Controllers. Installed Ubuntu Gnome 14.04. The cables I replaced were only those between my desktop machine and desktop router (D-Link DIR 655), but I believe I'm running Cat 5e (or 6) from inside the wall to my Netgear internet router. (I'd been working with Ubuntu 12.04 on my laptop from this same location using the same cabling and routers for a wired connection for more than 6 months, without incident or problem.) 

Didn't really notice any latency until I ran an update a day or two after my initial set up and noticed that the packages were lagging and some erroring out. But, merely inspecting the “edit connections” in Ubuntu, particularly the DHCP setting, resulted in a prompt completion of the update session, in the manner I've come to expect on my other Ubuntu based machines. 

But, the problem persisted. My internet access was very slow, and at times reported no connection. Pinging addresses of other machines on my intranet showed the lan ports were working just fine for this purpose, just over 0.2 ms per attempt with no packet loss and no intermittent slowing. (I'm working on a website that's currently hosted internally and I had this automatically booting in a browser tab with another browser tab pointing to an external site throughout the many reboot attempts involved in working on this problem. The internal site typically came up promptly and seconds faster than the external site (which was google.com).) Pinging sites on the internet resulted in times of 1500 or greater and, on at least two test attempts, all packets were loss. This corresponded to my visual experience using my web based mail server. Lots of “still sending” instances and sporadic loss of connection.

I tried various fixes reported by others as resolving what seemed like the same or similar slow performance of the RTL 8111/8168 lan ports (e.g., changed the driver from r8169 to r8168, disabled IPv6, assigned static IP ports – I thought it was an internal network problem for a while – and a half dozen or more other suggestions I came across in my searches, and, for what it's worth, I'm still running the r8168 module under the “it ain't broke” theory). It was only after signing up for email on this bug report that I decided to switch the existing CAT 5 cables between these lanports and the Dlink. Plugging in the first of the 5e cables resulted in immediate resolution of the internet connection problem (I even noticed the activity lights on the lanport flickering more quickly on the one with the 5e than the one without). It's been up and running fabulously for about 48 hours so, with no small amount of trepidation, I'm calling my particular iteration of this problem solved. I hope this helps others.