Add tested examples to documentation

[titusz]

Problem

Documentation code examples are not tested. When the API changes, examples can silently become outdated or broken, eroding user trust.

Goal

Ensure code examples in documentation are executable and verified by CI. This catches API drift and guarantees that users can copy-paste examples that actually work.

Current State

• 100+ code examples across 16 documentation files (tutorials, how-to guides, index page)
• Examples are well-written but untested
• Many tutorial examples are sequential (build state across multiple code blocks)
• Some examples involve file I/O (save/load/restore)
• Print output in comments uses # result style, not assertion-based

Considerations

• Sequential code blocks: Tutorials build state across blocks — need a strategy for self-contained vs. chained examples
• File I/O: Examples writing to disk need temp directory handling
• Non-deterministic output: HNSW results (key ordering, distances) may vary slightly
• MkDocs tab syntax: === "Tab name" indented code blocks may need special handling
• Tooling options: pytest-examples, doctest, plain pytest with extracted snippets, or other approaches — to be evaluated

Acceptance Criteria

• Documentation examples are executed and validated in CI
• Approach chosen balances test coverage with documentation readability
• No degradation of the current documentation quality