I see from the output that you have almost the same card that I have. You have the:
Realtek Semiconductor RTL8101E / RTL8102E PCI Express card
This is the Realteck webpage for the driver for Linux but you have to scroll down for Unix/Linux
This is the main page if you want to look further:
You said:" It's a fairly new computer. Here are my specs from All Settings >> Details in Linux Mint. I'm running Mint Maya 13 with Cinnamon right now.
Memory: 5.6 GiB
Processor: Intel Core i3 CPU M 380 @ 2.53GHz x 4
Graphics: Intel Ironlake Mobile
Disk: 239.7 GB"
That's plenty of Memory and with a single core you should be good to go and Mint 13 should be ok.
Install the driver and see if that helps.
There is a bug with your version of Mint and here is the bug I found:
Here is the Linux Mint website for tips after installation:
Here is the main page for your distribution:
While I was on the Mint site I looked for a Solved thread for your problem and didn't find one.
I'll keep looking to try to help you the best that I can.