It is assumed that the spectrum of electromagnetic radiation - and light is beside infrared, ultraviolet, X-rays or gamma radiation only a little part of is radiation - is continuous. Continuous means, that one can measure photons on all frequencies.
If electrons only absorb specific wavelengths of light and then emit them and that how we see color, but what happens to the other wavelengths?
Your confusion comes from replacing the fact how the identification of a chemical element through its absorption or emission spectrum takes place. Not the absolute frequencies of the excited states of the electrons is of interest, but only the relative distances to each other. This is obvious if some remember that the gas or solid during the measurement would have different temperatures. Sodium on the sun would be hotter than sodium on earth. You could identify this element only, if you compare the relative distance of the frequencies from this element.
Without them being absorbed ... but no light does get through a sufficiently thick material.
After absorption a re-emission happens. This can be as a reflection or - on very low frequencies (infrared) into the deep of the wall. (Read also about phonons, if you are interested.)