No, you cannot heat a warmer surface using a colder gas. You cannot reduce entropy. You are ignoring the 2nd law of thermodynamics again.
The surface does emit infrared light. This cools the surface. It takes energy to do that.
CO2, like pretty much any substance, does absorb infrared light and convert it to thermal energy. This does slightly warm the gas. Like any substance above absolute zero, it also converts thermal energy to electromagnetic energy (light) which also happens to be in the infrared range. It is a lower frequency, however. The gas is not as warm as the surface. It cannot heat the surface.
No atom or molecule will absorb a photon that has less energy than the atom or molecule already has.
CO2 just radiates light into space just like the surface. Just not as much light.
The practical upshot? The surface heating CO2 by radiant heat is just another way for the surface to be cooled by the atmosphere. The colder atmosphere CANNOT IN TURN HEAT THE SURFACE.