Creating own widgets for rqt

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Does it mean that now I have to create a project with Qt Designer and afterwards just copy the .ui file? (A project like based on an example Creating a Qt Widget Based Application ?)

No it's not mandatory. There's another rqt tutorial page that explains (sorry the link was missing from python tutorial you cited); in a nutshell, it's up to developers to use Qt Designer / .ui files or not. Although personally I highly recommend using it for better maintenance, if you choose not to use .ui files in your rqt plugin, there's still a portion of document about non-ui files.

But how will all the functions copy themselves to Python?

That tutorial should explain the idea too.

I managed to write some simple widget based on rqt_robot_steering (and add it to RosGui :)), but I had to write all callback functions by hand. Is there a way of doing it automagically?

In Qt in general you use Signal-Slot feature for callback mechanism, which is applicable to rqt as well.

Please add comments / modify your OP if something is still unclear.