(1, 2, 3) Though Linton's An outline of functorial semantics does contain the essence of the results and proofs of the monad–theory correspondence (see in particular Theorems 8.1 and 9.1), it is true that the terminology and style make it difficult to extract the results as we would expect to see them today. I think it is appropriate to cite this paper as the original reference, though citing the following papers for modern statements would be complementary.
Note that what we would now call a large algebraic theory, Linton calls a clone of operations.
(2, 3) As far as I can tell, the earliest reference in which the monad–theory correspondence appears in the modern form is Theorem III of Dubuc's Enriched semantics–structure (meta) adjointness. Dubuc establishes an equivalence between (large) $\mathcal V$-theories and $\mathcal V$-monads on $\mathcal V$, which in particular implies the classical result. The correspondence with all monads, rather than just finitary or $\kappa$-ary, seems to have fallen out of favour, and many modern treatments elide this case.
(1, 3) For the finitary version, the result (again in the enriched setting) appears as Theorems 4.3, 3.4, and 4.2 of Power's Enriched Lawvere theories. I haven't found an earlier reference with proofs.
(1, 2, 3) As far as I am aware, the only paper in which both results follow directly is Lucyshyn-Wright's Enriched algebraic theories and monads for a system of arities: in particular both the finitary and the large monad–theory correspondences follow from Theorems 11.8 and 11.14. Again, this all takes place in the $\mathcal V$-enriched setting.