The process from the console is more or less:
- create a folder to mount the device into e.g. "mkdir /usb"
- plug in the USB stick and issue the command "dmesg". You should see the plug in event in the last few lines. You are looking to see how the device is identified (probably sdb or sdb1)
- mount the USB stick into /usb "mount /dev/sdb1 /usb". Replace /dev/sdb1 with whatever you see in the dmesg command
- go into the stick "cd /usb"
- you can see what is there with a "ls" command
- install the driver. You can probably take a shortcut "rpm -ivh *.rpm" - saves typing.
- try loading the driver "modprobe e100"

In Linux you should unload the USB stick before removing it (or remove it with the server shut down).
cd /
umount /usb (or perhaps umount /dev/sdb1)
rmdir /usb

After plugging in the usb pen it may be worth checking if hotplugging mounts it automatically (I don't have a system to test). Do an "ls /mnt" and see if anything appears there. If it does, you can cd to it and install from there.

I have effectively zero Linux experience whatsoever. I will do my best to get the driver mounted by USB.

This is not so good if you have little Linux experience. RHEL, in EL7, removed a lot of old drivers from the distro. This affects CentOS7 and therefore ClearOS7. One of the drivers removed was the e100 driver for your NIC so it is not supported in ClearOS7. All is not lost, however. You could run ClearOS6 (probably not the best solution), or I have the kmod-e100 driver compiled here. You'll need to find a way of getting it onto your PC somehow. I don't think ClearOS7 auto-mounts USB drives but you can try, otherwise google how to mount a USB drive in Linux. Install the driver either with a "yum localinstall kmod-e100-3.5.24-1.clearos7.njh.x86_64.rpm", or, not so good, "rpm -ivh kmod-e100-3.5.24-1.clearos7.njh.x86_64.rpm". Then reboot the server or perhaps just do a "modprobe e100" and you should be OK.

Ethernet controller is Intel Corp NM10/ICH7 Family LAN Controller (rev 01)
Subsystem Dell Device 01ab

"External" is part of the initial setup of the NIC - the Role field. You can check with the command "cat /etc/clearos/network.conf", where your NIC should appear in the EXTIF parameter.

How do I check if the NIC is external? I am running the latest version that is available for download. I will run the command and let you know what the output is.

Presumably it is in standalone mode - no firewall. If it is, can you confirm that the NIC is set as external?
Which version of ClearOS are you installing?
Can you get to a command line from the console (alt+f2 or ctl+alt+f2)? If so, can you run the command "lspci -k | grep Eth -A 3"? What is the NIC type (the first line) and the driver in use?