You can get Gentoo up and running pretty quickly by following the handbook. From memory it's easy to miss one or two clear instructions because the styling of the handbook can add more eye-catching weight to the explanation than the actual commands. So be sure to re-read areas where things don't seem to working out.
Gentoo also has a binary repo if you don't plan to stray from whatever installation profile defaults you start off with.
I can't confirm a simple server install of Gentoo is somehow more lean than any other distribution.
I've used gentoo-install with success previously although I don't know how up to date it is.
I would try a few Plasma based tiling scripts before switching to anything like Sway or i3. You'll get a good idea of whether it's for you. Later on if you find you need more control over the tiling you could switch to a dedicated tiling window manager.
I'm using Karousel in Plasma which is scrollable tiling. You can install and enable it like so:
Go to System Settings > Apps & Windows > Window Management > KWin Scripts > select Get New... > In search enter Karousel and wait for it to show up > select Install > select the latest version (as of now karousel-0-9-4.tar.gz).
There is a companion desktop effect that also needs to be installed like so:
Go to System Settings > Apps & Windows > Window Management > Desktop Effects > select Get New... > enter Geometry Change in search > select Install > select the latest version (as of now kwin4-effect-geometry-change-1.3.tar.gz).
Karousel and Geometry Change have configuration options and Karousel also has keyboard shortcuts (view here) which you may need to update to your liking. I don't have a need to configure anything in Geometry Change as the default animation settings are fine. For Karousel I tend to adjust the various spacing and gaps options along with making sure the shortcuts I want are configured. That link above has a short video of what Karousel looks like in action.