The key to understanding how blackbody and absorption interplay is to remember that blackbody is a behavior that is found in the interaction between thermal and optical behaviors. It is not just one or the other. So, for instance, you will find that the nice clean concept of re-emitting at the same wavelength is more complicated because re-emission is just an optical behavior. In theory, an atom has to re-emit on the same wavelength (barring fluorescence) because it has to emit the same amount of energy as it received. However in a complex real body with thermal interactions some of that energy can be transferred into thermal energy. This allows a material to "absorb" light, converting it into heat. Likewise, the whole concept of blackbody is that thermal energy can be converted into light energy (photons).
When an object emits (via black body) and reflects (absorbing some colors), the effects are summed. However, it may be easier to first consider the case of an object that is emitting and absorbing transmitted light, like how stained glass absorbs some light and lets the rest through. We can consider: the sun.
This image is the optical spectra of our sun, courtesy of NOIRLab. In it we can see broad emission spectra (black-ish body) from the thermal energy of the sun, generating photons of all wavelengths. However, we see black bands. These bands are associated with the absorption of the light by the surface of the sun itself. The photons get emitted, but most photons of those frequencies collide with atoms that are in a state that can convert those photons into heat -- vibrations of the atoms in the sun.
Objects that are being lit from external sources and emitting via blackbody are similar to this example, being backlit. The energy is merely scattered on the surface of the object rather than passing through many layers of it. But the same rules apply. Light is absorbed on wavelengths where the atoms and their configuration can convert the light energy of that frequency into thermal energy, and emitted on all frequencies (and self-absorption limits the amount emitted on these absorbed frequencies).
From experience, I can tell you that part of the reason why this is a challenge to understand intuitively is that it isn't intuitive. Many years ago I got to work with a backyard aluminum foundry. Aluminum melts at 660C (1200F), which is high enough to glow visibly red, and starting to creep into orange as you get it hot enough to pour. This is an emission spectra -- it's similar to blackbody (nothing is really perfectly black body, but they get close). Thus, when poured into molds, it is glowing for a pretty long period of time before cooling to the point where we can no longer see it glow (still can burn your *@#%#ing fingers if you're a young foolish college student at the time!). Thus, during this time, the aluminum is both lit by incident light (from the sky) and is emitting.
When pouring ingots, I saw the most peculiar effect. It looked like I could see the solidified metal glowing deep in the heart of the ingot. This is, of course, foolish. The metal does not transmit -- it is opaque. But it felt like I could see light emanating from below the surface.
The scientist in me knows that what actually happened was that the corners of the ingot cooled first, so they began emitting less. The emitted light was concentrated on the center of the top of the ingot. However, with the reflected lighting, my brain got confused. We simply don't see objects that are emitting and reflecting at the same time. It isn't something we evolved to do, and it's not something I grew up with. So our brain, being the pattern matching engine it is, tries to determine the best match it can possibly make for the scene before it. My brain decided that the best match was a transparent material (like stained glass) with a light source on the inside radiating outward, so it ran with it. I was utterly confident I was seeing through the metal at an intuitive level, despite the scientist in me knowing exactly what the actual causes were.
So yes, it will be a bit counter-intuitive. You'll have to work out the math and physics first. Our visual centers are amazing creations, but they can indeed be fooled!