No Software Updates Are Available

You have this ppa installed ppa:langdalepl/gvfs-mtp.
It is used for mtp, but you should already have support in package gvfs-backends. There is no trusty support on the ppa, so you probably shouldn't be using it anyway. It might have been useful for Trisquel 6. Removing the ppa might fix your problem.

BugRep suggested: Removing the ppa might fix your problem.
Presumably that would be: gvfs-mtp

So I tried:
sudo apt-get remove gvfs-mtp
Reading package lists... Done
Building dependency tree
Reading state information... Done
E: Unable to locate package gvfs-mtp

That isn't it.

sudo apt-get remove mtp
Reading package lists... Done
Building dependency tree
Reading state information... Done
E: Unable to locate package mtp

That isn't it either.

sudo apt-get remove gvfs

Not the right approach. That removed bluefish, brasero, gnome-applets, trisquel-session, nautilus-share, gvfs-fuse, gvfs-backends, nautilus-sendto, and nautilus, which of course hosed my GUI, which I've repaired. After all that, and with gvfs gone, Software Updater's refrain is still, "Failed to Download Repository Information."

Looking at this PPA thing a little more: https://help.launchpad.net/Packaging/PPA

I'll guess this approach:

sudo apt-get remove ppa:langdalepl/gvfs-mtp
Reading package lists... Done
Building dependency tree
Reading state information... Done
E: Unable to locate package ppa

Nope, still no luck.

However, unchecking the URL's of two "other software" locations in Software Updater did get rid of the "Failed to Download Repository Information" result.

I have no idea how those two URL's got in there, except perhaps while I was attempting to transfer images from my Samsung SCH-U680 cell phone ...

And there's no need to update Kompozer, as it's no longer under development. And Terminal is happy again as well.