docs: Move code samples to files and other updates #116

Job	Run time
Lint check / Lint check (3.9)	20s
Unit tests / Unit tests (ubuntu-latest, 3.9)	23s
Lint check / Lint check (3.10)	20s
Unit tests / Unit tests (ubuntu-latest, 3.10)	23s
Lint check / Lint check (3.11)	24s
Unit tests / Unit tests (ubuntu-latest, 3.11)	21s
Docs check / Docs check	1m 42s
Lint check / Lint check (3.12)	23s
Async dostrings check / Async docstrings check	24s
Unit tests / Unit tests (ubuntu-latest, 3.12)	28s
Lint check / Lint check (3.13)	27s
Unit tests / Unit tests (ubuntu-latest, 3.13)	31s
Unit tests / Unit tests (windows-latest, 3.9)	1m 14s
Type check / Type check (3.9)	26s
Unit tests / Unit tests (windows-latest, 3.10)	2m 3s
Unit tests / Unit tests (windows-latest, 3.11)	2m 35s
Type check / Type check (3.10)	29s
Type check / Type check (3.11)	23s
Unit tests / Unit tests (windows-latest, 3.12)	53s
Type check / Type check (3.12)	25s
Unit tests / Unit tests (windows-latest, 3.13)	2m 0s
Type check / Type check (3.13)	33s
Integration tests / Integration tests (3.9)	24s
Integration tests / Integration tests (3.13)	33s
Integration tests / Approve integration tests for PRs from forks	0s
Integration tests / Integration tests for PRs from forks	0s
	18m 4s

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