I'm on the board (or whatever we are) of Spectra. I'll write to you offline to facilitate contact if you'd like to ask your question to our mailing list. I'm also posting this here in case there is anyone else who would like to get in touch with Spectra anonymously via a proxy: feel free to email me at "eriehl at math dot jhu dot edu".

"Taking the category of algebras" is a limit construction, more precisely analogous to "fixed points", but is there any reason to prefer this to the dual colimit construction, which yields the Kleisli category?

In your analogy "monads are categorified idempotents" you say "the analogue of taking the fixed points of an idempotent is taking the category of algebras of a monad." Here "taking fixed points" means splitting the idempotent, forming the equalizer of $m, 1_X \colon X \rightrightarrows X$, but the same object is recovered by instead taking the coequalizer of this pair (because, as you note, a splitting is a "direct summand": both a submodule and a quotient)... [to be continued]