If you transfer the heat to the atmosphere, it won't leave the Earth+atmosphere system, so the net effect will be zero. To actually cool the earth, you'd need to heat the atmosphere enough to make parts of it escape Earth's gravity.

Depends how high you send the heat, I would've thought...? If you ferry the heat above the current effective emission-to-space height (the mesosphere should suffice), you warm that high-up air and raise the effective emission-to-space height. Assuming a fixed lapse rate, a cooler surface temperature follows.