How to use the Editor to render a custom PsiElement tree

Please clarify what you need.

To display some text in Editor, you can do

editor.getDocument().setText(text)

I do not just show the text (not code), but also highlight, collapse, and click on it. How do I do that?

You can do custom highlighting using markup model:

editor.getMarkupModel().addRangeHighlighter(...)

foldling - using folding model:

editor.getFoldingModel().addFoldRegion(...)

Not sure what do you mean by 'click', but you can implement any logic in custom mouse listener added via

editor.addEditorMouseListener(...)

Thank you very much, can you help me with a problem:

https://intellij-support.jetbrains.com/hc/en-us/community/posts/115000710464-How-to-set-the-editor-s-background-and-overwrite-the-background-image 

Remove the editor's background image and caret row background in Popup?

To hide caret and caret row background in editor, you can do

((EditorEx)editor).setRendererMode(true)

As for disabling background image, I'll ask another person to respond, who knows better.