Interestingly, the UK and ROI are not in Schengen[1], but jointly have a Common Travel Area[2], which means Irish passport holders can live and work in the EU and in the UK, post Brexit. British and Irish citizens can generally live and work in the other country without visas.

[1] https://old.reddit.com/r/europe/comments/ms97x/venn_diagram_...

[2] https://en.wikipedia.org/wiki/Common_Travel_Area

No, the treaties which actually set the borders and acknowledge or cede sovereignty, for example the Paris Peace treaties of 1947 setting some of the post-WW2 borders, and many others, usually being peace treaties following some conflict that changed the borders.

Schengen is about regulating travel over the existing borders, Schengen is not a border treaty about where the exactly the borders lie and which country has full sovereignty over some land. While in colloquial language Schengen can be described as "eliminating borders", of course that is only a limited metaphor/analogy and the Schengen treaty does not challenge or change the validity of borders between countries in the Schengen area - even if it's trivial for people to cross over some Schengen border, it is not questioned that the sovereignty and laws change across that line.