Cursor color in the search bar
Cursor has a dark color with the dark theme. So it becomes almost invisible.
The main editor, console and output editors have a light cursor

This is on the Ubuntu system. On my Windows system, I have no such issue
I see that, but this may not be configurable in wxwidgets for native controls: https://forums.wxwidgets.org/viewtopic.php?t=39464
May be is it possible to switch to some other text control, say wxStyledTextCtrl, or even to the scintilla one, but in some restricted mode?
Technically yes, but it's a bit problematic. I think I already attempted that in the past, but there are still some challenges in limiting the size, as there is no proper way to limit the element size to one line.
Patches and proof of concepts are welcome ;). I can add ide:CreateTextCtrl method that creates wxTextCtrl by default, but you can patch it to create whatever you need (just need to make sure that it mimics the same API, like adding SetValue and ChangeValue methods that don't exist in wxStyledTextCtrl).