A version of the "transverse Calabi-Yau theorem" is proven by Kacimi-Alaoui in "Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications", Compositio 1990. I'm not sure how his "homologically orientable" condition compares to your "taut" condition. The main point of Kacimi-Alaoi is to develop the linear elliptic theory in the transverse setting, after which he says everything goes through.

You get a Fubini-Study metric once you fix an embedding in projective space; this requires not only an ample line bundle but a basis of global sections. There is no canonical way of choosing a basis, so no canonical Fubini-Study metric!

@TabesBridges This is roughly correct, but there is an infinite dimensional family of K\"ahler metrics in a fixed K\"ahler class, and the set of "Fubini-Study" metrics you describe is dense. So an ample line bundle is really very far from enough to produce a canonical K\"ahler metric!

I suppose, given a generator $C$, one can take the flat limit under each $\mathbb{C}^*$ in turn to obtain a new curve $\hat C$ which should be linearly equivalent and torus invariant.

It is not true that the automorphism group stays constant in flat families. For an explicit example, look at the degenerations of Fano threefolds with a two dimensional torus action to toric Fano threefolds (the papers of Ilten-Süss). For the construction of the moduli space, see the work of Odaka and Li-Wang-Xu.

I am not an expert, but it seems to me that it does (it follows from properties of the Hilb to Chow morphism, which is basically that flat convergence implies convergence as cycles). I will leave it to someone else to give a more definitive reference.

But it doesn't look to me like the answers given in the comments require reducedness of the limit?

How does this differ from your previous question? mathoverflow.net/questions/180933/…