Question 1
----------

I don't know if they are the *first* publications, but two very early references are
1.  Garrett Birkhoff (1940). *Lattice Theory*.
2. Øystein Ore (1944). "<a href="https://www.ams.org/journals/tran/1944-055-00/S0002-9947-1944-0010555-7/S0002-9947-1944-0010555-7.pdf">Galois Connexions</a>". (see Theorem 2 for a formulation of the "Corollary")

Remarks in these texts indicate that Galois connections between power set lattices $\mathcal P(X)$ and $\mathcal P(Y)$ induced by relations $R\subseteq X\times Y$ are due to Birkhoff, while Ore introduced the theory of Galois connections between arbitrary posets. From 2:
> It has already been pointed out by Garrett Birkhoff that any binary
relation defines a correspondence of the type of a Galois connexion between
the subsets of two sets

In the 1948 revised edition of 1, Birkhoff remarks in a footnote that the "Corollary" in the special case of Galois connections induced by relations is due to himself, while he refers to 2 as the origin of a particular axiomatization of Galois connections between arbitrary posets that he gives in the text.

As remarked by Marc Olschok in the comments, the "Theorem" occurs implicitly in

3. Saunders Mac Lane. *Categories for the Working Mathematician*.

in the form of an exercise. I don't have the first edition of that book to check whether this exercises already appears in the first edition. Given the similarity between adjunctions (due to Kan) and Galois connections, the generalization from the "Corollary" to the "Theorem" may well be a "folklore result", if not due to Mac Lane.