In terms of the greenhouse effect, what is the Earth's heat capacity?
For the greenhouse effect, the problem is not a matter of heat capacity. For example, Carbon Dioxide has a specific heat capacity of 0.843 J g$^{-1}$ K$^{-1}$ while $\ce{N_2}$, which is the largest component of the atmosphere, has a specific heat capacity of 1.039 J g$^{-1}$ K$^{-1}$ and water vapor, the most common greenhouse gas on earth, has a specific heat capacity of 1.865 J g$^{-1}$ K$^{-1}$. Thus, the warming isn't necessarily a thermodynamic problem, with regards to heat capacity (though it is, in a ways, a problem of entropy).

The reason that some gases are greenhouse gases is more of a radiative transfer problem than a thermodynamic problem, but the two overlay with regards to molecular structure. Water vapor's absorption spectrum shows that it absorbs most infrared radiation. This graphic demonstrates how much radiation of each wavelength is absorbed by different gases.

You may ask "well, that's dandy, but why do some gases absorb only some parts of the spectra?" Well, that is a bit more complicated, and is dependent on the molecular structure, quantum mechanics, and spectroscopy.

To answer your hypothetical questions directly:


  Has the earth’s atmosphere reached an equilibrium at which the input radiation energy roughly equals its output, or is the output significantly lower than the input because the energy is being used to heat up stuff?


In a ways, yes, that is a fundamental assumption, per Kirchoff's law.


  if the atmosphere’s composition remains exactly the same as it is now, will the atmosphere continue getting warmer, or will the temperature remain as it is now?


There would be a lag before major changes are seen, per Fourier's law. But since water vapor is a greenhouse gas, we would do best not to deus ex machina all greenhouse gases away. Additionally, we need greenhouse gases for survival- too little and we all freeze.


  Has the earth’s atmosphere reached an equilibrium at which the input
  radiation energy roughly equals its output, or is the output
  significantly lower than the input because the energy is being used to
  heat up stuff?


Well, we are almost always at some sort of radiative equalibrium, per Kirchoff's law. For example, the hotter an object becomes, the faster it will cool, per Stefan-Boltzmann Law.