So a little more info on this:
I was not able to translate any of the front page texts in this theme witm WPML at first. I contacted WPML and they told me to add the following code in the Custom XML Configuration:
<wpml-config>
<admin-texts>
<key name=”theme_mods_clean-enterprise-pro-child”></key>
</admin-texts>
</wpml-config>
After that I was able to find all the front page text in the string transaltions. My default site language is Swedish. All the string were defined as English. So I selected all the string and chose “Change the language of selected strings” to Swedish and everything worked. I had three languages (Finnish as the third), and everything was ok on the front page.
Except for the header image that was missing from the two languages that were not default (Swedish and Finnish).
So I added a header image and everything was fine.
Then I addes another header image, chose Randomize uploaded headers, and they were again gone from the two languages that were not the default languages (Swedish and Finnish).
BUT NOW, all the front page text strings have by themselves translated back to English! The flag still shows they are Swedish, but the content is English!
So I am pretty sure the theme broke my site.
Now, I can translate everything to Swedish again, but if I ever touch the header media, I will probably have to do everything all over again.
Surely this has got to be a whole bunch of bugs in your theme?