Class vs. function components and redux-toolkit vs "vanilla" redux are two independent decisions that don't have any impact on each other. (Though you should be aware that function components and hooks are recommended over class components for everything React).
How do the docs "stitch" the actions and selectors when using useDispatch and useSelector? Do that, but with the connect higher-order component instead.
The increment() function in the docs example that you posted doesn't just magically exist, it needs to be imported from the slice. You can export the entire actions object and use actions.increment but you usually see the actions exported as individual variables.