I uploaded a new version again. It should now display the list of pages in the correct language.
Yes it's fine now, when no menu is selected, its place is empty.
Are you viewing an English page when you change the the English menu?
No, the home page was opened in the Customize Theme screen, it is in Bulgarian. But I tried to go to the English home page (/en/) via the Language Switcher and then set the English menus. It didn't worked either.