Covariant formulation of physical equations? 
Is it possible to rewrite equations like the Klein-Gordon, the Dirac or the Proca equation in a generally covariant way? And if yes, how and how can the general covariance be shown?

(I searched for quite a while, but only was able to find contradictory statements. Some claim that general covariance is a defining feature of general relativity, but quite many claim this is incorrect. I think the easiest way to settle this would be showing explicitly how we can rewrite, for example, the Klein Gordon equation general covariant way.)
 A: Disclaimer: this is a brief answer, dealing with classical fields. To have a thorough understanding of the subject you should refer to books on GR and QFT in curved spacetime. A good read is the book General Relativity by Robert Wald.
You want to write down general covariant equations for some fields that reduce to the familiar ones (KG, Maxwell etc) when the metric is flat $g_{ab}=\eta_{ab}$.
One way to do this is the minimal substitution prescription:


*

*Physical quantities are described by the same kind of object, tensor fields (e.g. scalar field for KG);

*Equations of motion are modified by substituting the flat metric with the curved one $\eta_{ab}\rightarrow g_{ab}$, and partial derivatives with covariant derivatives $\partial_{a}\rightarrow \nabla_a$.


If you are familiar with gauge theories, in this way we are introducing a coupling to the gravitational field in the same way as when using minimal coupling to a gauge field.
Examples (your conventions may vary):
the Lorentz invariant Klein-Gordon equation 
$$ \partial_a \partial^a \phi - m^2 \phi = 0$$
becomes the generally covariant
$$ \nabla_a \nabla^a \phi - m^2 \phi =g^{ab}\nabla_a \nabla_b \phi - m^2 \phi= 0.$$
Maxwell's equations become
$$ \nabla^a F_{ab}=-4 \pi j_b $$
$$ \nabla_{[a}F_{bc]}=0.$$
At this point it should not be hard to convince yourself that you go back to the original form of the equations if the spacetime is Minkowski.
Things get slightly more complicated with the Dirac equation, and you need to consider spinors in curved spacetime and the spin connection.
Minimal substitution also works directly on actions/Lagrangians, as long as care is taken in dealing with the integration measure/tensor densities.
I emphasise that minimal substitution is one way to do what you want: you could clearly add to the equations terms that couple the fields to the curvature in a more complicated way, as long as it has the right covariance properties (e.g. you could add a curvature-dependent "mass" term to the KG equation), and you may have good reasons to do so.
