I think you need to do several things here; this is what I did when I had a similar problem (see the three commands below). I removed vlc and all its configuration files by running the first command; with the second I removed all the extra items it installs, and with the third I cleaned the apt cache so the package needs to be downloaded again; i.e. /var/cache/apt/archives is cleared.

sudo apt-get purge vlc

sudo apt-get autoremove

sudo apt-get clean 

As has been suggested, it is also important to remove any traces of the program from your home folder, as the purge procedure does not remove them, so remove the following folders as well (use -i instead of -f if you want to confirm removal). It is alright to remove the vlc folders here and not just the contents of the folder, as the folders will be reconstituted when you reinstall vlc.

rm -rf ~/.cache/vlc

rm -rf ~/.config/vlc

rm -rf ~/.local/share/vlc 

That should help eliminate the problems you encountered with the configuration of vlc; it is particularly important to remove the vlc folder in .config as that contains the file vlc-qt-interface.conf, which you mentioned appeared in the error logs and seems to be preventing vlc from opening.

Now you should be able to run

sudo apt-get update  

sudo apt-get install vlc

This should sort out the problem, but we can look into other possibilities if necessary and if you provide further information.