I don’t think this is carbon capture in the sense that most people mean: you still had to expend a lot of energy to fell the tree and process it into cardboard. Putting it somewhere where it won’t decompose will probably only prevent a tiny percentage of the carbon required to produce it. Thus the goal of cutting down fewer trees, and thus reusing and recycling.
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.