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The torus $T^2$ is the quotient of ${\mathbf R}^2$ by ${\mathbf Z}^2$. I denote by $z = [x,y]$ the class of $(x,y) \in {\mathbf R}^2$. I am used to denote $\Delta_\alpha \subset T^2$ what you denote b …

I think there are some theorems which are easier to prove in the diffeological framework, or as you say: for which the proof reveals more conceptual reasons. For example this one ? Proposition Let $X …

I would add to the previous list of "applications" of diffeology in ordinary differential geometry the theorem about Homotopic invariance of De Rham cohomology (§6.88 of Diffeology). In brief: Theorem …

Perhaps the general solution to your question can be found in "A theory of plots" by Atsushi Yamaguchi here? Slides from his talk at the last conference on diffeology and differential homotopy are her …