What is the theory of non-linear forms (as contrasted to the theory of differential forms)?

The answer to "what kinds of things can you integrate" depends on the context.

• Measurable functions are things you can integrate over measure spaces, which includes in particular measurable subsets of R^n.
• Differential forms are things you can integrate over oriented smooth manifolds -- the key thing about them is that their integrals are invariant under smooth, orientation-preserving changes of coordinates.
• Densities are things that can be integrated in a coordinate-independent way on any smooth manifold, regardless of whether it has an orientation or not.
• Coming full circle, every Riemannian manifold (i.e., smooth manifold endowed with a Riemannian metric) has a naturally-defined density dV, so in that context you can integrate measurable functions again: the integral of the function f is defined to be the integral of the density f dV.

All three of the expressions you asked about are examples of densities. For details, see my book Introduction to Smooth Manifolds, pp. 375-382.

In my opinion, you're looking for the notion of a cogerm.

If I understand correctly, the fact that such things act on paths (and not just vectors) allows for "higher order" forms like $d^2 x$, and the fact that such things aren't assumed linear allows for "non-linear" forms like $ds := \sqrt{dx^2+dy^2}$. And yes, there is indeed a notion of integration for such forms; see the link.