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.