CI: Run specs for vendored gem gitlab-housekeeper

What does this MR do and why?

This MR adjusts the CI configuration to run child pipeline for gems/gitlab-housekeeper.

It also fixes specs and rubocop CI jobs.

From !144421 (comment 1772010578):

So it seems we never configured this to run in CI so I created !144676 (merged) . I knew I remember adding this but it was actually added in the first POC MR !134485 (closed) (diffs) but then I extracted that to a new MR which was merged but I forgot to add the CI part.

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