C-algebras: See many things in the folder Noncomm geom and Cstar-alg. Same for Noncomm geom in the sense of Connes, Cyclic homology
Loday: Cyclic homology. In Noncomm geom folder. Covers many aspects of Cyclic and Hochschild homology. Among many other topics: Secondary char classes (section 11.5), Homology of small categories (App C), periodic and negative cyclic homology, Andre-Quillen homology, Deligne cohomology. For the latter, the main point is that there a cyclic homology complex of Connes which computes integral coeffs reduced Deligne cohomology but which has strictly commutative products!! This is stated for smooth algebras over C, not sure if it can be generalized to more general schemes. Also I am not sure if this has any relevance for non-reduced Deligne cohomology.
nLab page on Noncommutative geometry II