"Shininess" is basically a property that is based on the proportion of specular reflection to diffuse reflection. Specular reflection basically means the directions of parallel rays relative to each other are preserved, while diffuse means they are disrupted. This is because what "shine" really is is an image (a virtual image, more precisely) of the light source (perhaps distorted), and thus the rays coming out must "look like they came from a light source" meaning that their geometric relationships must be suitably preserved so as to present the same pattern to the eye as a light source of that shape and behind the object would present its rays to the eye. Since the only real light source is the one shining on the object, this means the reflection must suitably preserve the relative geometry of the rays as they reached it having come from the light source illuminating.
This is not due to the reflectivity of materials, rather it is due to the smoothness of their surface. If you rough up metal, it won't be shiny. The rougher it is, the less "shine" it will have as the rays will be more disturbed from their original configuration when impinging upon the surface. Smooth plastic is shiny as well. Smooth coal is shiny! Look at these pieces of lignite - lowest grade of coal - that has been smoothed into nice spheres:
http://www.mineralminers.com/html/jetsphs.stm
Cloth will never be shiny because as a fibrous material, its surface is innately irregular due to the fibers. Matte paint forms a rough microstructure on its surface when it dries - see:
https://en.wikipedia.org/wiki/Paint_sheen#Technology
(sorry I don't have source better than WP at the moment to keep things quick - if you can find one though, please suggest it.)
You also asked another question about how impinging EM waves cause charges to move when they "don't contain any charges". The answer to this is that charges don't respond to other charges, they respond to EM fields. The force law is $F = qE$ (yes I'm ignoring magnetic fields for simplicity), force is charge times electric field, not charge times another charge. Coulomb's law is not the most general force expression, and you should think of it as not $F = k\frac{Qq}{r^2}$ but rather $F = q\left(k \frac{Q}{r^2}\right)$ where the parenthesized term is an electric field produced by the "master" charge $Q$ and the charge of interest $q$ (which is which is arbitrary) is experiencing a force due to $Q$'s field. It is thus by producing an EM field that charges "respond to other charges", but they only directly respond to the field itself (which is very important to understand if you want to understand EM waves, since the same principle also applies to changes in the field: a change in the field only affects the field immediately next to that, not immediately a distant charge, and thus a wave propagates.). Since an EM wave is composed of electric and magnetic fields, when it impinges upon charges they will react to it because, as said, they respond to fields.
Best Answer
In quantum mechanics light is emergent from zillions of photons, and photons as quantum mechanical particles have interactions with the spill field of the lattice that composes metals and (all other solids). It so happens that in metals the optical and lower wavelengths have a high probability of being elastically scattered and thus reflected, that is why metal surfaces make good mirrors. Keeping to classical electrodynamics, this is recorded in the reflectivity of metals.
"The metal absorbs the microwaves", not in a simple but a more complicated manner Classical elecrodynamics gives a simpler explanation for how the grid functions:
As the two ways of studying electromagnetism are consistent to each other, the classical framework is easier to understand than the quantum mechanical. It would involve interference effects of photons in a convoluted manner that the classical avoids. See this to understand about how interference effects appear with single photons at a time. In general classical electrodynamics is much simpler because it mathematically describes the behavior of photons with matter in a simpler way.