Wasn't the CO2 in the cardboard once in the atmosphere and absorbed by the tree when it was alive? So the net CO2 is the same, in the same carbon cycle.
Probably the same with methane: enters the atmosphere, gets decomposed in H2O and other stuff that go back to Earth and are still part of the same cycle.
Methane does ultimately turn back to CO2, but until it does it's a much more powerful greenhouse gas than CO2. In any case, the decomposition in the landfill aborts the carbon storage argument, as the carbon is released back into the atmosphere.
Probably the same with methane: enters the atmosphere, gets decomposed in H2O and other stuff that go back to Earth and are still part of the same cycle.