Thank you for posting information about the issue with downshifting speed and slowing down your server response. I am aware of an issue where sometimes the network connection negotiated 10 Mbps link after waking up from sleep. A driver update came out with a fix last year.
With the built-in network connections the BIOS has an affect. Are you using the latest BIOS for the desktop board? Another thing you might try is doing a complete uninstall of the Intel Network Connections software including the base driver.
Just to make sure the files are gone after the installation, search for e1k62x64.* files and delete them. Then install the latest software from scratch.
Let me know if any of this helps. If not, I will do my best to help find a solution.