Dear David,
Yes, models of single-sorted algebraic theories are always monadic over Set, and such theories correspond precisely to finitary, ie omega-filtered-colimit-preserving monads on Set. If we take the correspondence
between single-sorted algebraic theories and Lawvere theories for granted, this is stated eg in Hyland--Power [1], with further references there. More generally, monads preserving alpha-filtered colimits for a higher regular cardinal alpha correspond to algebraic
theories with alpha-ary operations; unbounded monads (such as the powerset monad) can be viewed as corresponding to "large theories" with no bound on their arities. The theory corresponding to the powerset monad, eg, is the theory of sup-lattices, ie posets
with arbitrary small joins, which is not expressible with operations of bounded arities. Regarding references on these generalizations, I would also be curious.
Concerning your questions on extensions of theories, and more general kinds of theories and base categories, there recently was a long thread on the category theory zulip server [2], which I'll try to summarize:
extensions of single-sorted theories by new operations and/or equations are always monadic (regardless of arities); this follows from the fact that monadic functors have the left cancellation property (Proposition 3.3 in [3]). Extensions by new sorts are typically
not monadic, eg Set x Set is not monadic over Set. The models of many-sorted theories are monadic over powers of Set, and extensions of many-sorted theories by operations and axioms are also monadic, again by cancellation. Things become more complicated in
the generalized/essential algebraic case, since (in the generalized algebraic, ie dependently typed case), adding new operations can create new sorts by substitution, which can lead to successive monadic extensions which are not composable, as Tom Hirschowitz,
James Deikun, and possibly others pointed out. In general there's a lot of ongoing work on the dependently typed, ie generalized algebraic case, such as eg Chaitanya Leena Subramaniam's recent PhD thesis representing dependent algebraic theories by finitary
monads on presheaf categories over direct categories [4].