Thank you guys very much for help, but still confused. Server was installed long time ago before me, probably the users is messed up. It is plesk. Just spoke with godaddy support:
"Okay. Most of the time this happens because you have PHP support running in Apache mode instead of FastCGI.
If you're in Apache mode, any changes take root ownership.
Unfortunately this is just how Plesk is built. If you want to be able to access within Plesk you need to have it as FastCGI mode.
You'd need to ask the developers of Plesk that question, I don't know why it acts this way, I just know that's what happens.
If you want to be able to have access to all files in Plesk without permission issues, you're going to need to run in FastCGI mode. I don't know of any work around for that."
So he is telling me that I MUST to use FastCGI instead of Apache. But I do not understand how so many people using Prestashop on Apache without any problems at all.
So I have 2 questions: 1) How I can fix 500 internal server error in PrestaShop in FastCGI mode
OR
2) How can I fix user permission issue under Apache mode?
Looking right now about SuPHP, is it any different way to fix it?
Found one more way to fix it: http://www.prestashop.com/forums/topic/226561-solution-for-recursive-write-permissions-on-files-and-folders-error/ Is it safe and correct way to fix that problem?
Thanks.