You are welcome... again...
1.) Okay, after downloading Linux Mint Mate version 18.3 and installing that to a USB stick (16gb) and running the typical new installation updates, I downloaded the current qTox AppImage file "qTox-v1.16.3.AppImage" from their link which I updated in my previous reply, installed "ffmpeg" version 3.x, and tried to double-click the AppImage file from the file manager, and I got the same "shared library" error message you did? I did not have this problem except in Linux Mint Mate 18.x
Current qTox AppImages
So, I opened a console terminal in the home folder where I put the AppImage (/apps), and ran the typical command for AppImages below (just add ./ to the front of the file name), and it came right up.
Then, I thought I would create a desktop shortcut launcher to the AppImage, but that would not work in Linux Mint Mate 18.x? So, I remembered there was a companion application called "AppImageLauncher" that is available from the link below for all editions and versions of Linux Mint 64-bit and 32-bit, there are specific ones for Linux Mint 19.x based on Ubuntu 18.04 Bionic, and I installed the regular 64-bit deb file, restarted Mate, then I could double-click the AppImage file and it now asked If I wanted to run it once which works now, or integrate it with my Mate desktop, pretty cool. The strange thing is that although it added it to my Mate menus, I still could not launch qTox AppImage from the menu, but If I right-clicked the menu item for qTox and added it to my desktop, or my system tray panel, those worked perfectly?
https://github.com/TheAssassin/AppImage ... r/releases
64-bit deb file for Linux Mint 18.x and 17.x
https://github.com/TheAssassin/AppImage ... _amd64.deb
32-bit deb file for Linux Mint 18.x and 17.x
https://github.com/TheAssassin/AppImage ... b_i386.deb
2.) As for my qTox installation script in Linux Mint Mate 18.3, I have not yet tested that because I did not want to "skew" my fresh Mate 18.3 testing environment for software available as AppImages by installing the qTox source code and dependencies, but I will report on that. FYI: I meant for the qTox installation script to be run from people's Home folder.
3.) Remember there is also the "Flatpak" option which can be installed from the Software Manager, but I have not tried that.
Hope this helps ...