Mine was to work tax policy to incentivize companies to make all their packaging shiny and white, incentivize people to litter, and disincentivize everybody from recycling.
My friend's was to use a giant rocket to push the earth farther away from the sun
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. Aside from problematic effects on weather, this would be really hard because the upper atmosphere is very thin and so has low heat capacity and low heat conductance.
I was thinking about direct-radiation heatsinks: the inner part made of super-heat-conductive material that transports heat from ground level to a huge radiator fan in outer space, insulated by an outer layer on the way up (otherwise you lose all the heat to the atmosphere). But it would have to be both superconducting and very, very large.
Also, the cooling effect would be localized at the bottom end, so what would you stick it into? A volcano?
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.