OK, so first things first. I forgot to mention that your copying the output was not entirely pointless, it proved that you are using the correct repositories and are indeed experiencing connection issues.

/ is your root directory ("folder").
/etc/hosts is a file. You can dump it from the terminal by typing...Or, to be able to access it from your PC...
SSH is a communication protocol for logging into a Linux machine remotely. It exposes the command-line interface ("shell"). Indeed, SSH stands for "secure shell" since it uses the public key infrastructure.. You need another Linux machine to use SSH, or something like PuTTY on Windows. It might be a bit involved if this is completely new to you but it is worth having a go and learning a new skill.

As I mentioned, another way of copying the results of a terminal command is redirecting the output into a file. This is done the same way in Linux as in MS-DOS (actually, dating back to CP/M times). The generic format is...
Doing that will redirect the output from the screen to file.txt. You will not see anything on the screen until the command execution has finished and it returns back to the prompt, all the output will be in the file. You can then grab the file in USB mass storage mode and copy it here.
To see the file in mass storage mode you need to create it somewhere under MyDocs. So, for example, run...
Once done, attach the phone to the USB in mass storage mode (make sure it was not attached while you ran the above command!) and you should see a file apt.txt in your phone's root directory.

Regarding your connection/router issues, could you try connecting to TMO and/or updating somewhere else? Take your phone to a cafe with free WiFi and try there.