Some further thoughts: the most striking results I know of on "purely algebraic cyclic/Hochschild homology" are due to Wodzicki, see e.g.

*Wodzicki, Mariusz*, **Homological properties of rings of functional-analytic type**, Proceedings of the National Academy of Sciences USA 87, No. 13, 4910-4911 (1990). ZBL0717.46063

which states that stable $C^*$-algebras have trivial cyclic homology. Obviously this doesn't answer your $II_1$ factor question...

Also: your remark that *in some cases, we can ignore the analysis and make the situation a bit simpler* confuses me a little. To get anywhere with cyclic or Hochschild homology, we need to do some kind of comparison of resolutions, or construction of contracting homotopies, or something like that. My intuition — but I don't work much on operator algebras, so I could well be wrong here — is that a von Neumann algebra is such a big object we usually can only get a handle on it by looking at suitable subsets which generate its unit ball in the WOT/SOT. So for group von Neumann algebras, one tries to see what's going on for translations, and thence to deduce more general results by exploiting $w^*$-$w^*$ continuity; or else use projections and approximation arguments. If we go to a purely algebraic category, then it is no longer sufficient to define things on dense subsets — one really needs a global definition, one really needs to verify that certain putative identities are satisfied by each element of the von Neumann algebra.

Sorry if that's a bit waffly. I think my point is that imposing continuity restrictions actually makes things *easier*, because — intuitively — more things are going to be projective/injective/flat relative to one's restricted class of short exact sequences. This is why, for instance, we know that $H^n_{cb}(M,M)=0$ for any von Neumann algebra $M$, but why the analogous claim without the '$cb$' is open and back-breaking. In a similar vein, if you work in a restricted category then one does indeed get some known instances of homological non-triviality (though at the level of modules, not at the level of cyclic homology):

*Polyakov, M. E.*, **An example of a spatially nonflat von Neumann algebra**, J. Math. Sci., New York 113, No. 2, 350-359 (2003). ZBL1042.46031.

I should also say that the Hilbert module stuff you mention doesn't really connect to your original question about cyclic (co)homology. It's interesting, and I think more has been done, but it's just *different* — so if that's what interests you, cyclic and Hochschild homology may be something of a distraction.