What is the duration and manner of the pause between the two Friday sermons? Is the admonition within them obligatory, or is the sermon valid without it? And where does it stand among the pillars?

It is sunnah to sit lightly between the two Friday sermons, for a duration equivalent to reciting Surat Al-Ikhlas. This sitting is not obligatory.

Admonition (Maw'izah) is a fundamental pillar of the sermon, and the sermon is not valid without it. Its minimum is enjoining the fear of Allah (taqwa). It is recommended that the admonition be the last of the pillars.