I switch between different types of main window view and my torrents disappear. But in left part (category) they are and the traffic IN and OUT is moving. I even downloaded some but can't see it in Tixati. How to switch it in visible mode?  http://i.imgur.com/jlQ9XaZ.png

Double check that textbox that there is no blank space in there, because if there was it would filter out torrents without it/them.

Most likely it's the Status Filter. All the buttons on the right of the Text Filter are turned off. To display everything, you should enable all of them.

Ah yes Pete, you are right missed that, not used to using them myself.