Make API requests work

[smnandre]

• Route API request commands through the Playwright server
• Preserve response status, headers, repeated headers, and binary-safe bodies
• Share cookies with browser contexts and support storage state, disposal, and tracing
• Add focused unit and real-browser functional coverage

Tests:

• PHP CS Fixer
• PHPStan
• PHPUnit: 1697 tests, 3740 assertions, 3 skipped

[codecov]

Codecov Report

❌ Patch coverage is 90.62500% with 3 lines in your changes missing coverage. Please review.

Files with missing lines	Patch %	Lines
src/API/APIResponse.php	87.50%	3 Missing ⚠️

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