Dependency problems when updating gcc-4.9-base (and similar) packages

I normally check for updates myself, just to be sure that I have the latest everything. When I went to check for updates again today, nothing appeared in the list! So I click Install Now anyway. Software Updater gives me the following error:

Package dependencies cannot be resolved.
This error could be caused by required additional software packages which are missing or not installable. Furthermore there could be a conflict between software packages which are not allowed to be installed at the same time.

I also got a red icon on my Ubuntu panel that displays this message:

An error occurred, please run Package Manager from the right-click menu or apt-get in a terminal to see what's wrong. The error message was: 'Error: Marking the upgrade (E:Error, pkgProblemResolver::Resolve generated breaks, this may be caused by held packages.)'. This usually means that your installed packages have unmet dependencies.

. So I opened Synaptic, and clicked Mark All Upgrades. It wants to get rid of at least a hundred different packages (plenty of which are critical for Ubuntu to work!) to install just a handful! When I mark them anyway, I get:

E: Error, pkgProblemResolver::Resolve generated breaks, this may be caused by held packages.

W: Ignoring file 'getdeb.list.bck' in directory '/etc/apt/sources.list.d/' as it has an invalid filename extension
W: Ignoring file 'getdeb.list.bck' in directory '/etc/apt/sources.list.d/' as it has an invalid filename extension

I decided to click Edit > Fix Broken Packages, but upon doing so I get:

E: Error, pkgProblemResolver::Resolve generated breaks, this may be caused by held packages.
E: Unable to correct dependencies

Packages gvfs-daemons and libstdc++6 are marked as broken, but I don't want to have to remove so many packages just to resolve dependencies. What should I do?!?

Output of sudo apt-get install -f:

Reading package lists... Done
Building dependency tree       
Reading state information... Done
0 upgraded, 0 newly installed, 0 to remove and 10 not upgraded.
N: Ignoring file 'getdeb.list.bck' in directory '/etc/apt/sources.list.d/' as it has an invalid filename extension
N: Ignoring file 'getdeb.list.bck' in directory '/etc/apt/sources.list.d/' as it has an invalid filename extension

Output of sudo apt-get install --reinstall gvfs-daemons libstdc++6:

Reading package lists... Done
Building dependency tree       
Reading state information... Done
0 upgraded, 0 newly installed, 3 reinstalled, 0 to remove and 10 not upgraded.
Need to get 660 kB of archives.
After this operation, 0 B of additional disk space will be used.
Do you want to continue? [Y/n] y
Get:1 http://archive.ubuntu.com/ubuntu/ trusty-updates/main gvfs-daemons amd64 1.20.3-0ubuntu1.2 [108 kB]
Get:2 http://ppa.launchpad.net/dolphin-emu/gcc-for-dolphin/ubuntu/ trusty/main libstdc++6 amd64 4.9.2-0ubuntu1~14.04 [269 kB]
Get:3 http://ppa.launchpad.net/dolphin-emu/gcc-for-dolphin/ubuntu/ trusty/main libstdc++6 i386 4.9.2-0ubuntu1~14.04 [283 kB]
Fetched 660 kB in 10s (60.1 kB/s)                                              
(Reading database ... 1325331 files and directories currently installed.)
Preparing to unpack .../libstdc++6_4.9.2-0ubuntu1~14.04_amd64.deb ...
Unpacking libstdc++6:amd64 (4.9.2-0ubuntu1~14.04) over (4.9.2-0ubuntu1~14.04) ...
Preparing to unpack .../libstdc++6_4.9.2-0ubuntu1~14.04_i386.deb ...
Unpacking libstdc++6:i386 (4.9.2-0ubuntu1~14.04) over (4.9.2-0ubuntu1~14.04) ...
Setting up libstdc++6:amd64 (4.9.2-0ubuntu1~14.04) ...
Setting up libstdc++6:i386 (4.9.2-0ubuntu1~14.04) ...
Processing triggers for libc-bin (2.19-0ubuntu6.6) ...
(Reading database ... 1325331 files and directories currently installed.)
Preparing to unpack .../gvfs-daemons_1.20.3-0ubuntu1.2_amd64.deb ...
Unpacking gvfs-daemons (1.20.3-0ubuntu1.2) over (1.20.3-0ubuntu1.2) ...
Setting up gvfs-daemons (1.20.3-0ubuntu1.2) ...
N: Ignoring file 'getdeb.list.bck' in directory '/etc/apt/sources.list.d/' as it has an invalid filename extension
N: Ignoring file 'getdeb.list.bck' in directory '/etc/apt/sources.list.d/' as it has an invalid filename extension

Output of apt-cache policy gvfs-daemons x11-utils libstdc++6 gcc-4.9-base:

gvfs-daemons:
  Installed: 1.20.3-0ubuntu1.2
  Candidate: 1.20.3-0ubuntu1.2
  Version table:
 *** 1.20.3-0ubuntu1.2 0
        500 archive.ubuntu.com/ubuntu trusty-updates/main amd64 Packages
        100 /var/lib/dpkg/status
     1.20.1-1ubuntu1 0
        500 archive.ubuntu.com/ubuntu trusty/main amd64 Packages
x11-utils:
  Installed: 7.7+1
  Candidate: 7.7+1
  Version table:
 *** 7.7+1 0
        500 archive.ubuntu.com/ubuntu trusty/main amd64 Packages
        100 /var/lib/dpkg/status
libstdc++6:
  Installed: 4.9.2-0ubuntu1~14.04
  Candidate: 4.9.2-0ubuntu1~14.04
  Version table:
 *** 4.9.2-0ubuntu1~14.04 0
        500 ppa.launchpad.net/dolphin-emu/… trusty/main amd64 Packages
        100 /var/lib/dpkg/status
     4.8.4-2ubuntu1~14.04 0
        500 archive.ubuntu.com/ubuntu trusty-updates/main amd64 Packages
     4.8.2-19ubuntu1 0
        500 archive.ubuntu.com/ubuntu trusty/main amd64 Packages
gcc-4.9-base:
  Installed: 4.9.2-0ubuntu1~14.04
  Candidate: 4.9.3-0ubuntu4
  Version table:
     4.9.3-0ubuntu4 0
        500 archive.ubuntu.com/ubuntu trusty-proposed/main amd64 Packages
 *** 4.9.2-0ubuntu1~14.04 0
        500 ppa.launchpad.net/dolphin-emu/… trusty/main amd64 Packages
        100 /var/lib/dpkg/status
     4.9.1-0ubuntu1 0
        500 archive.ubuntu.com/ubuntu trusty-updates/main amd64 Packages
     4.9-20140406-0ubuntu1 0
        500 archive.ubuntu.com/ubuntu trusty/main amd64 Packages
N: Ignoring file 'getdeb.list.bck' in directory '/etc/apt/sources.list.d/' as it has an invalid filename extension
N: Ignoring file 'getdeb.list.bck' in directory '/etc/apt/sources.list.d/' as it has an invalid filename extension

UPDATE: The problem has returned. As previously suggested, the trusty-proposed source is still unchecked, but the very same problem has now appeared in the main channel (trusty-updates). I need to resolve the dependencies, or else I will not be able to easily update everything. I have the information stored in the Ubuntu Pastebin.

As you can see in the output of apt-cache policy gvfs-daemons x11-utils libstdc++6 gcc-4.9-base, the installation candidate for gcc-4.9-base is 4.9.3-0ubuntu4:

gcc-4.9-base:
  Installed: 4.9.2-0ubuntu1~14.04
  Candidate: 4.9.3-0ubuntu4
  Version table:
     4.9.3-0ubuntu4 0
        500 archive.ubuntu.com/ubuntu trusty-proposed/main amd64 Packages
 *** 4.9.2-0ubuntu1~14.04 0
        500 ppa.launchpad.net/dolphin-emu/… trusty/main amd64 Packages
        100 /var/lib/dpkg/status
     4.9.1-0ubuntu1 0
        500 archive.ubuntu.com/ubuntu trusty-updates/main amd64 Packages
     4.9-20140406-0ubuntu1 0
        500 archive.ubuntu.com/ubuntu trusty/main amd64 Packages

This version is available in trusty-proposed.

But the problem is, libstdc++6 has version 4.9.2-0ubuntu1~14.04 and there is no candidate with the version 4.9.3-0ubuntu4 to install and the versions of libstdc++6 and gcc-4.9-base have to be exactly the same.

libstdc++6:
  Installed: 4.9.2-0ubuntu1~14.04
  Candidate: 4.9.2-0ubuntu1~14.04
  Version table:
 *** 4.9.2-0ubuntu1~14.04 0
        500 ppa.launchpad.net/dolphin-emu/… trusty/main amd64 Packages
        100 /var/lib/dpkg/status
     4.8.4-2ubuntu1~14.04 0
        500 archive.ubuntu.com/ubuntu trusty-updates/main amd64 Packages
     4.8.2-19ubuntu1 0
        500 archive.ubuntu.com/ubuntu trusty/main amd64 Packages

For this reason, disable trusty-proposed:

sudo sed -i '/trusty-proposed/d' /etc/apt/sources.list

and repair

sudo apt-get update
sudo apt-get install -f

Update for the update in your question:

Upgrade

sudo apt-get install gcc-4.9-base=4.9.3-0ubuntu4

Downgrade

sudo apt-get install libstdc++6=4.8.4-2ubuntu1~14.04

Install

sudo apt-get install gcc-4.8-base