When Task Manager exe is runned, it appears for a second and then disappears also from the system tray at the right bottom of the desktop

See this:

http://windowsxp.mvps.org/ToolsQuit.htm

Also please follow the instructions HERE & then post your log in a new thread in the Spyware, Adware, Viruses and HijackThis Logs section.
(Not in this section please).