click "W" the it open Window Ink.
Sounds like your Win- is "stuck". Press and release the Win- key. If that is not enough you can start the On-Screen Keyboard (osk.exe) and it will show you that the key is pressed down. Click the key to turn that off.
Who knows why your symptom happens the way it does. Sounds like a coincidence. Otherwise you might guess that there is some software associated with the HDMI which does that. Note that it would be convenient for you to press just K or P then (instead of having to press Win-K or Win-P) to see the Connect and Project dialogs. Or you could press just Ctrl+Enter (instead of Win+Ctrl+Enter) to start Narrator and perhaps hear a reason for the "feature".
FYI
Robert Aldwinckle