docs: overhaul documentation structure and build

Author: lavr

.github/workflows/tests.yaml

@@ -72,10 +72,14 @@ jobs:
           cache: pip
       - name: install dependencies
         run: |
-          pip install -e .
+          pip install -e ".[html]"
           pip install sphinx -r docs/requirements.txt
       - name: build docs
         run: sphinx-build -W -b html docs docs/_build/html
+      - name: run doctests
+        run: sphinx-build -b doctest docs docs/_build/doctest
+      - name: check links
+        run: sphinx-build -b linkcheck docs docs/_build/linkcheck
 
   typecheck:
     name: "typecheck"