Beginner's guide to band structure follows. I've taken considerable liberties with the details to simplify this so don't take it too literally!
This is going to seem an odd place to start, but consider filling up the atomic orbitals in an atom with electrons. If you take a noble gas, e.g. Xenon, you'll find each orbital is filled completely with two electrons and this is why Xenon is inert. If you take Potassium instead you find all the lower orbitals are filled with two electrons, but the outmost orbital contains only one electron so the orbital is only half full. This is why Potassium is very reactive.
When you clump atoms together into a solid, the interactions between the atoms spread the sharp atomic orbitals into energy bands. Suppose our solid contains $n$ Xenon atoms, then each band can contain 2$n$ electrons. But each Xenon atom contributes 2 electrons to each band, so the energy bands in solid Xenon are all full. That's why solid Xenon is an insulator. In the case of Potassium all the lower energy bands are full up with 2$n$ electrons, but the top band contains only $n$ electrons, i.e. it is only half full, because each Potassium atom has only 1 electron left over to put into this band. That's why solid Potassium is a conductor.
The position of an electron in an energy band doesn't just determine its energy, it also determines its momentum. If you want to make an electron move so it can conduct electricity you need to change it's momentum, and therefore you need to change its position in the energy band. But when bands are full you cannot change an electron's energy/momentum because there are no free spaces in the band for the electron to move into. That's why filled bands are insulating and part filled bands are conducting.
Now, if you imagine taking your solid and filling up the energy bands with electrons there is going to be a highest occupied band and a lowest unoccupied band. Now the nomenclature can be a little confusing. If the highest occupied band is full (like solid Xenon) we tend to refer to it as the valence band, and the lowest unoccupied band as the conduction band. The energy difference between the bands is the band gap. The reason why we call the lowest unoccupied band the conduction band is because any electrons that get excited into it will conduct; electrons in the valence band won't conduct (because the valence band is full).
But, if the highest occupied band is only part full (like solid Potassium) we call this band the conduction band because the electrons in it can conduct. Strictly speaking the highest band is both the valence band and conduction band, but convention dictates we call it the conduction band. In metals we're usually not fussed about the lowest unoccupied band and the band gap because they aren't involved in conduction of electricity.
Now, on to transparency. When a photon interacts with an electron it transfers it's momentum to the electron i.e. it changes the momentum of the electron. But if you recall from above, you can't change the momentum of an electron in a full band. The only way to change the electron momentum is to hit it hard enough, i.e. with enough energy, to make it jump over the band gap into the lowest unoccupied energy band. So if you measure the optical absorption as a function of energy you find there's little absorption until the photon energy matches the band gap, and the absorption suddenly rises. For many materials the band gap energy corresponds to ultra-violet light, so the solid doesn't absorb visible light i.e. it's transparent. As you say, these solids are also insulators because the same mechanism (change of electron momentum) determines both conductivity and optical absorption.
In metals the lowest occupied band (the conduction band) is only partially full so electron momentum can be changed by any amount you want. That's why metals absorb light (and radio waves etc) very strongly and are opaque.
Incidentally you do get borderline cases. Pure silicon is an insulator, but the band gap is only about 1.12 eV and this is less than the wavelength of red light. So silicon absorbs light even though it's an insulator. Well, it's an insulator in the dark. As soon as you shine light on it the electrons you excite over the band gap conduct electricity, so silicon conducts when you shine light on it.
I hope all this helps. If you want to clarify any of the above please comment.
This is a fantastic question, and a topic I was very confused about when I first took a class on Radiative Processes. The ultimate answer, as hinted at by @LubošMotl, is anything---if you start with a 'white-noise' of radiation (i.e. equal amounts of every frequency), it will equilibrate with the medium/material into a black-body distribution because of its thermal properties (see: Kirchhoff's Law, and the Einstein Coefficients). This is just like if you gave each molecule in a gas the same energy, they would settle to a Boltzmann Distribution.
In practice (and hopefully a more satisfying answer) is that it's generally a combination of line-emission and Bremsstrahlung, with Bremsstrahlung1 dominating at high temperatures ( $T \gtrsim 10^6 -10^7 K$ ). Lines are produced at myriads of frequencies depending on the substance of interest, and the thermodynamic properties (e.g. temperature). For everyday objects, I think the emission is primarily from molecular-vibrational lines. Individual lines are spread out by numerous thermodynamic broadening effects to cover larger portions of the spectrum. Finally, as per Kirchhoff's law, equilibrated objects can only emit up to the black-body spectrum. In practice, you'll still see emission/absorption lines imprinted, and additional sources of radiation.
Lets look at a breakdown of the relevant transitions as a function of energy level:
radio: nuclear magnetic energy levels (also cyclotron emission in the presence of moderate magnetic fields).
microwave: rotational energy levels
infrared: vibrational energy levels (molecules)
visible: electronic (especially outer electron transitions)
ultraviolet: electronic (especially outer/valence electron ejection/combination)
x-ray: electronic (inner electron transitions)
gamma-ray: nuclear transitions
1: Bremsstrahlung (German for 'braking radiation') is radiation produced by the acceleration of charged particles---most often electrons. This can happen between any combination of bound (in atoms) or unbound (free or in plasma) charges.
Best Answer
X-ray (and gamma rays) are quite penetrating. They can pass through solid matter with much less attenuation than visible light as an example.
But that doesn't mean that the attenuation is zero. Put enough "stuff" in the way, and the energy is eventually scattered or absorbed. In the case of the atmosphere, it's "just" air, but there is quite a bit of it. The depth of the atmosphere is plenty to stop almost all UV/X/gamma radiation.
In fact most types of EM radiation are blocked by the atmosphere. But our eyes see only the transparency in visible light.
The small molecules that make up most of the atmosphere ($N_2$, $O_2$, $Ar$) take a lot of energy to excite. It turns out that visible light is just shy of the energy to do this efficiently, so interactions are very rare. More energetic forms (including X-rays) can ionize these molecules, absorbing or scattering the radiation. Given a thick enough layer, almost all the incoming radiation is removed.