first is konqueror. When I launch it says:
The requested operation could not be completed
Cannot Initiate the Protocol
Technical Reason: Unable to Launch Process
Details of the Request:
URL: start.fedoraproject.org
Date and Time: Fri February 27 2009 22:10
Additional Information: Unable to create io-slave: klauncher said: Unknown protocol ''.
Description:
The program on your computer which provides access to the protocol could not be started. This is usually due to technical reasons.
Possible Causes:
The program which provides compatibility with this protocol may not have been updated with your last update of KDE. This can cause the program to be incompatible with the current version and thus not start.
You may have encountered a bug in the program.

But than I click new tab in menu, click home and it shows me http://start.fedoraproject.org/ just fine. Whats the matter, maybe I can fix it somehow.
Another question is that kate doesn,t want to save it's settings when I close it. Why?

For Kate try checking the contents of $KDEHOME/share/config/katerc and its permissions.
Try checking for a typo in the Home URL, especially in the config file ( $KDEHOME/share/config/konquerorrc )

Whay! I am stupid. I just forgot to add http:// to my home page address. Than why home button worked in new tab? ha-ha. As for kate it saves everything but in new session. On start it just loads default which I didn't modify. I have just to set my modified session as default ))) Sorry for bothering