Do gases reflect some IR radiation?

This kind of pictures is often misleading. When a paradox appears, common sense tells us that there is very likely something wrong with the assumptions. Here the bad assumption is to display the whole atmosphere as a single box what leads to an apparent paradox.

In reality the density varies with altitude and this has for consequence that the mean free path of photons varies with altitude too. Variations of all radiative parameters with altitude follow.

To see the difference with a picture where the atmosphere is represented by N layers, let us take an example for N=2. We suppose that a solid surface S is radiating F (W/m²) into an atmosphere constituted of 2 layers L1 and L2 and the system is in a steady state. We suppose farther that L1 absorbs all the radiation coming from S.

L1 must reemit all the absorbed radiation (steady state) and because of isotropy, F/2 is reemitted up and F/2 is reemitted down. A part P of the upgoing F/2 will be absorbed by the upper layer L2 and of course again reemitted P/2 up and out of the system and P/2 down. As this downgoing P/2 will be absorbed by L1 and again reemitted P/4 down and P/4 up, it is proven that the flux going from the atmosphere to the surface S is F/2 + P/4 + ... > F/2. so it is actually no paradox that the downgoing flux is more than a half of the flux emitted up by the surface. The values depend on the detailed radiation properties of the layers.

Of course N=2 is still not realistic enough and I have omitted the algebra that can be done by any interested reader but the purpose was to show that an N layer atmosphere doesn't behave like a 1 layer atmosphere. Among others I have omitted the source of energy that is necessary to make up for the difference between F that S emits and F/2 + ... etc that S absorbs from the atmosphere.

Regarding the reflection, there is none in this particular case. The infrared radiation emitted by the surface between 0°C and 30°C is mostly absorbed by the vibrational spectrum of H20 molecules (partly by CO2 too). So this radiation is indeed absorbed and exactly reemitted with the exception of a small window that goes directly to space. Obviously there is always some scattering (clouds) but the processes in the dense and warm lower atmosphere are dominated by absorption and emission in the IR spectrum.