I am wondering: Are there any general theorems or principles relating the theory of Z-graded dg objects and the theory of Z/2Z-graded dg objects? I am mainly interested in dg algebras, dg Lie algebras, and dg categories over fields of characteristic zero.
To me, the main difference between the Z-graded and Z/2-graded cases is that the former allows certain simplifying boundedness restrictions, which in the latter do not seem to make sense. Typically, one considers a nonpositive (in the cohomological grading) Z-graded dg-algebra and dg-modules bounded above or below over it, as appropriate. The case of connected, simply connected (in the sense of cochains, not just cohomology) nonnegative dg-algebra is similar.
The typical simplification achieved under such restrictions is that a dg-module whose underlying graded module is projective is always homotopy projective. Also, the two ways of defining the differential derived functors (by taking infinite direct sums or products along the diagonals) become equivalent, since the sums/products are actually finite.
When one has to consider dg-algebras that do not satisfy the above kind of restrictions and/or unbounded dg-modules, the Z-graded situation is not any simpler than, and not much different from, the Z/2-graded situation.
References: 1. Keller "Deriving DG-categories"; 2. Husemoller, Moore, Stasheff "Differential homological algebra and homogeneous spaces".