Very similar issue with #4 . It occurs when I restart the system, when I log out of the system and log back in, or when I disable and re-enable the extension through gnome-tweaks. The extension settings pane still shows the time zone I set, but the prayer times shown are as if they were the default (i.e. GMT +00:00). This is fixed when I go into settings again, change them to something else, and then change them back. Using the following bash command from a terminal also fixes the issue, but it has to be run every time the issue occurs:
dconf write /org/gnome/shell/extensions/azan/timezone "'X'"
Where X is to be replaced with the desired time zone number.

I have downloaded the extension "azan@faissal.bensefia.id" through Ubuntu Software, in Ubuntu 18.04.