docs: enhance framework examples and SQL playground

[cofin]

Summary

• Fix framework documentation with correct driver types and extension_config patterns
• Add multi-database examples for FastAPI, Flask, and Starlette showing async+sync database configurations
• Enhance SQL playground with sortable datatable, blinking cursor, improved table header styling
• Fix sphinx_datatables extension error by ensuring _static directory exists before build
• Update Litestar dependency injection docs with DuckDB advanced configuration patterns

Changes

Framework Examples

• FastAPI: Fixed type annotations, removed from __future__ import annotations (causes FastAPI inspection issues), added multi-database example
• Flask: Added factory pattern with init_app(), added multi-database example
• Starlette: Clarified session access patterns, added multi-database example
• Litestar: Enhanced dependency injection docs with proper extension_config usage and DuckDB hooks

SQL Playground

• Added sortable columns with sticky headers
• Added blinking golden cursor for visibility
• Improved table header styling (light/dark mode aware)
• Combined SQL statements using execute_script()
• Moved status indicator to left of buttons

Build Fixes

• Fixed sphinx_datatables extension error by creating _static directory before the extension writes to it

Test plan

• All framework example tests pass (pytest docs/examples/frameworks/)
• Documentation builds successfully (make docs)
• Pre-commit hooks pass