All of what you mentioned makes good sense and thanks for the links.
When a computer is shut down and turned off but the cable modem is left on, is the computer still connected to the Internet?
If yes and someone continues to use XP after the support end date, should they unplug their modem after shutting off and turning off the computer?
The computer isn't connected to the Internet anymore, no. But it's advisable to disconnect it completely by either switching off the power supply if there's a switch available on the back of it, or to remove the power cord. By doing that, you'll be performing
a so called 'cold boot' every time you power on. See here for more details: http://www.reference.com/motif/society/difference-between-cold-boot-and-warm-boot
One other thing since you're using Firefox. If you have the view with the orange, oblong Firefox button top left, hit ALT to toggle the text links on and then go to Tools | Options | Privacy,
Next, checkmark the option to "Clear history when Firefox closes" and then click the "Settings" button. In the submenu which appears, checkmark "Cache", but remove the checkmark from the others. Here's a couple of screenshots.


There's malware around which downloads to the browser cache and then sits there waiting for the moment when you inadvertently land on a compromised site and then activates. By clearing the cache every time you close Firefox, anything in there will be deleted
automatically.