Can be that problem because I installed ES launcher with vrtec.montessori.si support and then install ES via directory on vrtec.montessori.org ? So ES packages was withouth supported domain?
Nope, it will not affected this.
Look at that
[13-Jan-2017 22:13:38 UTC] PHP Warning: passthru() expects at least 1 parameter, 0 given in /home/ki130820/public_html/test.php on line 3
Error_log
It seems like this error log latest error was recorded at 12 Jan 2017, mean when we perform Easysocial update from your site, it should generate a similar error message on your server error log file but that is not.
I suspect other issue causing this installation issue, can you request with your webhosting provider temporary turn off those configuration setting which can override the real PHP fatal error from your server? Because now your server always return 500 internal server error and that was difficult for us troubleshoot this installation issue.
Once your webhosting provider turn it off, I will investigate again.