I've tried this /tmp folder trick and it didn't work for me; what did though is this:
In .htaccess file in /sites/default/files folder i've changed this:
Options -Indexes -ExecCGI -Includes -MultiViews
to this
#Options -Indexes -ExecCGI -Includes -MultiViews
Options +SymLinksIfOwnerMatch
Have no idea if it has a security issue or anything else, since i'm total zero in this things, but it worked :)
Well, I also have all the folders in /sites/default/files 775'd, including files itself.
UPDATE #1!
Actually, only commenting the first line like this
#Options -Indexes -ExecCGI -Includes -MultiViews
works for me, so one of those options standing on our way to image styles. I don't know exactly which one, please feel free to play with them.
UPDATE #2!
Actually, the update #1 works only in conjunction with the following setting in the hosting panel for my domain:
Use Apache instead Nginx for handling the static files like .jpg .gif .js .css .rar .exe .mp3 etc.
Yes, I have this setting in the hosting panel.
So, seems like it's final - the two things - commenting Options in the .htaccess and use Apache instead of Nginx