Explicit procedure for integrating densities (twisted $n$-forms)? I'm aware of several slightly different (yet equivalent) definitions for the density bundle of a non-orientable manifold. Unfortunately, I've been unable to make sense of integration in any of them. I'll phrase the question in terms of the definition which I like the most. Feel free to answer with a different definition if simplifies things.
Let $X$ be a manifold. Let $\mathcal{OR}$ be the orientation sheaf on $X$ which gives for every open set $U \subset X$ the abelian group $\mathcal{OR}(U) = H_n(X,X-U ; \mathbb{Z})$.
Definition: The sheaf of compactly supported $n$-forms twisted by orientation is called the sheaf of densities and is denoted by $\mathcal{S =  \Omega^{n}_{c} \otimes_{\mathbb{Z}}\mathcal{OR} }$ .
Question 1: Let $\omega \in \Gamma (U,\mathcal{S})$ be a local section. How do I get/compute the integral $\int_U \omega $ out of this information?
 A: The key to answering your questions in my opinion is to get a nice description of densities in local coordinates. (For this purpose, the definition of densities that you have chosen does not seem very convenient to me). The point is that, as for $n$-forms, one can describe the restriction of a density to the domain of a coordinate chart by a single smooth function. However, under a coordinate change, the function corresponding to a density transforms by multiplication with the absolute value of the Jacobian of the chart change (in constrast to $n$-forms, where just the Jacobian of the chart change enters). Once you have this description, you can define integration exactly as for $n$-forms in the oriented chase. 
The simplest way to get to this description is to define the bundle of densities as an associated bundle to the linear frame bundle corresponding to the represenation defined by the absolute value of the determinant. Then the above description holds by definition. To get the description in your picture, you probably just have to check that the action of a diffeomorphism on the orientation shaef you define is given by multiplication by the signature of the Jacobian (which perfectly fits the intuitive meaning of this sheaf).   
