Fix: Replace deprecated docker-run-action with manual Docker setup

[JosueNina]

Description

Replace addnab/docker-run-action@v3 with manual docker run to fix Docker client version error (1.41 too old, needs 1.44)

• Remove deprecated action
• Add docker helper function
• Run container manually with tail -f
• Execute all commands via runInContainer

Related Issue

Closes #9283

Motivation and Context

N/A

Requires Documentation Change

N/A

Types of changes

• Bug fix (non-breaking change which fixes an issue)
• Refactor (non-breaking change which improves implementation)
• Performance (non-breaking change which improves performance. Please add associated performance test and results)
• New feature (non-breaking change which adds functionality)
• Breaking change (fix or feature that would cause existing functionality to change)
• Non-functional change (xml comments/documentation/etc)

Checklist:

• My code follows the code style of this project.
• I have read the CONTRIBUTING document.
• I have added tests to cover my changes.
• All new and existing tests passed.
• My branch follows the naming convention bug-<issue#>-<description> or feature-<issue#>-<description>

[Martin-Molinero]

Thank you!