I do not understand what is your resquest. Here is what I observed during when browsing your site.
You have 5 different home pages for 5 different languages. Switching from one to the other seems to work well except that you did not translate the content (which is always in English). So it appears that you have 5 homepages with the same content (even if you assigned them a different language).
When browsing the English version, I did not find a page which has a translation (maybe some exist but I did not find them).
It seems also that you use the same menu for all languages so only the English version can be browsed.
The only translation I found is the tagline (only on the german homepage).
The theme options or Settings->Reading are interlinked. Changing the homepage will set it on both places. If you provide me with an email, i’ll send you a download link to the theme for test purpose (couldn’t find one on your profile).