Investigate and fix Docker 29 API compatibility issues

Summary

We've had a number of bugs related to Docker 29 lately. On digging further, I found that Docker 29 introduced breaking changes that impact Runner functionality. This issue tracks investigation and remediation of Runner's compatibility with Docker 29.

Release notes: https://docs.docker.com/engine/release-notes/29/.

Problem

Docker 29 raised the minimum supported API version from 1.41 to 1.44 (released Nov 10, 2025). This broke multiple Runner components that hardcoded or assumed API 1.41 support.

Known Impact Areas

• GitLab Runner build failures for Docker deploym... (#39100)
• Volume Maintenance fails with Docker Version 29 (#39101 - closed)
• error "clear-docker-cache.sh" : Error response ... (#39117 - closed)

Acceptance Criteria

• Audit Runner codebase for hardcoded Docker API version references
• Review Docker 29 release notes for additional breaking changes affecting Runner
• Determine API version negotiation strategy
• Assess backward compatibility requirements (minimum supported Docker version)
• Document supported Docker versions in Runner docs
• Create follow-up issues for any required fixes