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What you are searching is called tiling window manager, there are several with various features.
On Gnome there is a hotkey for horizontal side-by-side windows, on Plasma there are more layout options and keyboard shortcuts, but I'm not sure if any of them has one-third tiling.
https://help.gnome.org/users/gnome-help/stable/shell-windows-tiled.html.en
There was an app or extension that let's you use the number pad as directional arrows for window placement. Like 7 would move to the top left corner, 4 would move it as half screen to the left. Hitting them again does resizing. I'm trying to remember the name, it wasn't an lntuitively named tool. But you could try Awesome Tiles to see if that works.
I know you said Gnome, but if you are willing to look at Plasma, I've just started using Bismuth on KDE Plasma and I think it can do at least a chunk of that. It can set particular sizes with Window Rules, it looks to have a quite robust shortcut system, including resizing windows, swapping, rotating, or changing layouts. As for the focus vs open, KRunner lets you choose the active application when you type it's name. There's also this: https://github.com/academo/ww-run-raise but I have not used it and cannot vouch for that.
Seems like Bismuth is unmaintained. Polonium seems to be its successor.
I will check out Polonium! Thanks!
i3wm is built for keyboard control, though I am slightly confused about your usecase so take that recommendation with a grain of salt.