Are we changing Earth's weather with global warming?
First, regarding your assumptions, here is a graphic from this online textbook. 



As you can see, your first assumption is correct, surface radiation is much more significant than thermals. Your second assumption is also correct, 'Back Radiation' and 'Emitted by Atmosphere' flows are significant, but absorption of heat by the ground from the air isn't even featured on this chart. So conductive heat transfer from air to ground is negligible. 

This page gives some good background on how carbon dioxide contributes to global warming, and, as you said, it is due to absorption of infra-red radiation from both the ground and the air. Specifically, CO$_2$ absorbs IR wavelengths that can 'escape' absorption in water vapor (the primary greenhouse gas) in the range 12-15 $\mu$m, as shown in the graphic below.



With more thermal radiation absorbed in the air, less escapes into space, and the net energy balance of Earth is positive which results in a slow and steady warming.