ci: prepare workflows for GitHub merge queue

[serrrfirat]

Summary

• add merge_group support to the core CI workflows used by PR A
• remove staging-specific skips and cache-save conditions from test.yml and code_style.yml
• fire Claude review on active non-draft PRs instead of the old staging-promotion label

Validation

• parsed the updated workflow YAML locally with PyYAML
• ran git diff --check
• did a focused workflow-semantics review on the final diff

Not in this PR

• deleting staging promotion infrastructure
• changing regression-test-check.yml for merge_group
• docker / release-plz / staging branch cleanup

Part of #2719.

[gemini-code-assist]

Note

Gemini is unable to generate a review for this pull request due to the file types involved not being currently supported.

[henrypark133]

Closing this as superseded by #3104. The branch strategy changed: main already contains staging, so the replacement PR handles the main-only workflow cutover directly instead of preparing staging for a later promotion.