How the heck can I install Qt5 on Ubuntu 16.04 LTS?

I have loads of programs that require Qt4 and I need to install programs that require Qt5. When I try to install Qt5 things start off dodgy and then proceed to insanity. I request friendly guidance. It's probably easiest for you to look at the terminal output directly:

$ sudo apt update
Ign:1 http://linux.dropbox.com/ubuntu wily InRelease
Ign:2 http://repo.ubports.com xenial InRelease                                                                  
Hit:3 http://linux.dropbox.com/ubuntu wily Release                                                              
Hit:4 http://gb.archive.ubuntu.com/ubuntu xenial InRelease                                                      
Ign:5 http://repo.ubports.com xenial_-_mir26 InRelease                                                          
Hit:6 http://ppa.launchpad.net/deadsnakes/ppa/ubuntu xenial InRelease                                           
Hit:8 http://gb.archive.ubuntu.com/ubuntu xenial-updates InRelease                                              
Hit:9 http://repo.ubports.com xenial Release                                                                    
Hit:10 http://security.ubuntu.com/ubuntu xenial-security InRelease                                              
Hit:11 http://gb.archive.ubuntu.com/ubuntu xenial-backports InRelease                                           
Hit:12 http://repo.ubports.com xenial_-_mir26 Release                                          
Hit:13 http://ppa.launchpad.net/maarten-baert/simplescreenrecorder/ubuntu xenial InRelease
Hit:14 https://repo.nordvpn.com/deb/nordvpn/debian stable InRelease          
Hit:15 http://ppa.launchpad.net/nextcloud-devs/client/ubuntu xenial InRelease
Hit:17 http://ppa.launchpad.net/numix/ppa/ubuntu xenial InRelease             
Hit:19 http://ppa.launchpad.net/otto-kesselgulasch/gimp/ubuntu xenial InRelease
Hit:20 http://ppa.launchpad.net/ubuntu-toolchain-r/test/ubuntu xenial InRelease
Hit:21 https://updates.signal.org/desktop/apt xenial InRelease
Hit:22 https://riot.im/packages/debian xenial InRelease
Reading package lists... Done 
Building dependency tree       
Reading state information... Done
23 packages can be upgraded. Run 'apt list --upgradable' to see them.

$ sudo apt install qt5-default
Reading package lists... Done
Building dependency tree       
Reading state information... Done
You might want to run 'apt-get -f install' to correct these:
The following packages have unmet dependencies.
 protonmail-bridge : Depends: libqt5designer5 but it is not going to be installed
                     Depends: libqt5multimediawidgets5 but it is not going to be installed
 qt5-default : Depends: qtbase5-dev
               Depends: qtchooser (>= 55-gc9562a1-1~)
E: Unmet dependencies. Try 'apt-get -f install' with no packages (or specify a solution).

$ sudo apt install qtbase5-dev
Reading package lists... Done
Building dependency tree       
Reading state information... Done
You might want to run 'apt-get -f install' to correct these:
The following packages have unmet dependencies.
 protonmail-bridge : Depends: qt5-default but it is not going to be installed
                     Depends: libqt5designer5 but it is not going to be installed
                     Depends: libqt5multimediawidgets5 but it is not going to be installed
 qtbase5-dev : Depends: libgles2-mesa-dev but it is not going to be installed or
                        libgles2-dev but it is not installable
               Depends: libqt5concurrent5 (= 5.9.5+dfsg-0ubports2) but it is not going to be installed
               Depends: qt5-qmake (= 5.9.5+dfsg-0ubports2)
               Depends: qtbase5-dev-tools (= 5.9.5+dfsg-0ubports2)
               Depends: qtchooser
               Recommends: libqt5opengl5-dev (= 5.9.5+dfsg-0ubports2)
E: Unmet dependencies. Try 'apt-get -f install' with no packages (or specify a solution).

$ sudo apt-get -f install
Reading package lists... Done
Building dependency tree       
Reading state information... Done
Correcting dependencies... Done
The following packages were automatically installed and are no longer required:
  audacity-data libflac++6v5 liblilv-0-0 libportsmf0v5 libqscintilla2-l10n libqt4-declarative libqt4-network
  libqt4-script libqt4-sql libqt4-sql-mysql libqt4-sql-sqlite libqt4-xml libqt4-xmlpatterns libqtcore4
  libqtdbus4 libqtgui4 libsbsms10 libserd-0-0 libsord-0-0 libsratom-0-0 libunibreak1 libvamp-hostsdk3v5
  libzlcore-data libzlcore0.13 libzltext-data libzltext0.13 linux-headers-4.15.0-42
  linux-headers-4.15.0-42-generic linux-image-4.15.0-42-generic linux-modules-4.15.0-42-generic
  linux-modules-extra-4.15.0-42-generic simplescreenrecorder-lib sqliteman-doc
Use 'sudo apt autoremove' to remove them.
The following packages will be REMOVED
  audacity fbreader libqscintilla2-12v5 libqt4-designer libqt4-opengl libqt4-svg libsuil-0-0 libzlui-qt4
  protonmail-bridge simplescreenrecorder sqliteman vidyodesktop
0 to upgrade, 0 to newly install, 12 to remove and 23 not to upgrade.
1 not fully installed or removed.
After this operation, 45.6 MB disk space will be freed.
Do you want to continue? [Y/n] n

The final suggestion of my attempt would remove some core things I need, so this is not an option. How might I proceed, kindly stranger?

clean up and update:

sudo apt-get autoremove && sudo apt update && sudo apt upgrade

Download Qt5:

wget http://download.qt.io/official_releases/qt/5.7/5.7.0/qt-opensource-linux-x64-5.7.0.run

Install:

chmod +x qt-opensource-linux-x64-5.7.0.run \
./qt-opensource-linux-x64-5.7.0.run

Install g++:

sudo apt-get install build-essential

Install generic font configuration library - runtime:

sudo apt-get install libfontconfig1

Configure a compiler:

Launch Qt Creator. Go to Tools > Options. Click Build & Run and select tab Kit. Configure a compiler if it is not automatically detected.

Install OpenGL libraries:

sudo apt-get install mesa-common-dev

Source:

https://wiki.qt.io/Install_Qt_5_on_Ubuntu