How to execute python code before blender operator is executed Lets say there are operators (buttons .etc) that run or save the text in Text Editor. I need to change the text before such operators execute.

Is there a way involving some callback functions, listeners or handlers?

If there is not, is there a way to remove UI element and replace it in the same spot with custom operator? I only found a way how to append or prepend an operator to such menu or header.
I don't think that there are listeners which you can use in that way. 

But what you can do is your idea in the second question. You could replace the button with a custom operator which executes your code first and calls the original operator afterwards.
You can access the sources if you right-click on that original button and choose edit source. After that go to the scripting view and choose the source file available in the dropdown box