From the question and the audio settings posted in the pastes from the commands
list-modules there may be an issue with a pulseaudio module to restore an output device. Let me suggest further steps to narrow doen the issue:
Temporarily remove modules
We don not know if that helps but we may be able to narrow the issue down if we temporarily removed modules that are in question.
To remove a module issue the following in a terminal:
pacmd unload-module <index>
<index> will be listed from
pacmd list-modules. Try to unload any single one of the following loaded modules (by index!) to test if it is the culprit:
- module-suspend-on-idle (because your issue is associated on idle of Skype)
- module-loopback (because you don't need the "Monitor of..." device)
- module-switch-on-port-available (unlikely but worth a try)
- module-intended-roles (switches to the intended device when a phone call comes in)
The default setting with loading of the default modules will be restored by running
Check for errors
Also you may want to see if there are any errors from pulseaudio in the system log files before your HDMI sink is connected.
Remove user-space configuration
Sometimes we have erroneous configuration settings in the user-specific settings. To overcome this me may try if deleting
~/.pulse helps (it will be recreated on next start of pulseaudio)
mv ~/.pulse ~/.pulse.old
pulseaudio -k # to restart pulseaudio
Generate a pulseaudio log
Another approach would be to check for useful messages to narrow down the issue by generating a pulseaudio log file from running pulseaudio in verbous mode (see here on how to do this).
Let us know of the result to better point you to a permanent solution.