Published posts not showing up & updated page not updated

Using WordPress 3.6.1.

1)
Posts are published and they appear in the list of posts as published. But they do not show up on the main blog page. When I insert the link of one of the new posts in a browser, the post appears and correctly in “Recently Updated” list all the new posts. Once I go back to the main page, they are not there. Any tips?

2)
Updated page does not get updated. While I am logged in as admin, I can see it correctly displayed, but once I log out, the old page is showing up.

I am not a geek so please while helping out, try to be as specific as possible in the steps I should take.

Thanks a lot!

web (in Czech):
http://ism-czech.org/

This is the most recently published post and it also appears as first on the list of posts. However, not showing up on the main page:
http://ism-czech.org/2013/10/06/jeruzalem-neni-hlavnim-mestem-izraele/