Oh, good question, how to make Kate work well under GNOME. I have to admit, I use it under KDE, so never really dealt with the theming. But I believe, “Tokyo Night” is only the editor theme. Can you select a different Window Color scheme in the menubar under Settings?

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what’s the difference between what looks like three different folder tree buttons (Document seems to only show one file, and then Project and File Browser plugin both show the full tree of the folder you have opened)?

• “Documents” only shows your currently open tabs. To be honest, I never quite figured out what it’s good for, but I think it makes more sense, if you use Kate for authoring texts or such. I normally disable it in the settings, under Plugins → Documents Tree.
• “Filesystem” is kind of like a mini-file-manager. You can navigate to any directory you want in there, or have it always show the current folder of the document you have currently open. But it isn’t aware of what a repository is, so depending on how you open Kate, it may not show the right folder and jumping to the current document’s folder will put you into a sub-directory of your repository. As I said above, I also mostly keep that one disabled these days, although I can see it being useful.
• “Projects” is aware of Git. It always shows the current repository folder, if you are in one, expanding the file tree from there. It hides files listed in .gitignore. And yeah, in my opinion just what you want to use for programming.

And is there an equivalent for the “Code Runner” plugin? If not, I guess I could always just run “python filename.py”, but a play/run button would be nice.

There is a plugin called “Build & Run”, which you can enable and which might do what you’re looking for. I typically prefer running from the terminal, so I can’t say too much about it…