1) Yes. In fact, free oxygen exchange is a requirement for plant health.

2) They are about the same. The only advantage to the cardboard is that it will breakdown in time. But it can create anaerobic conditions until that happens.

Myth of Paper Based Mulches

Myth of Landscape Farbric

Perhaps because the cardboard retains water once it gets wet? I know that worms love cardboard covered areas with a burning passion, so there must be some kind of oxygen coming through.

A key aspect of landscape fabric’s affect on soil that I observed here when I briefly used it on paths 20 years ago is not addressed in the link(s) gg48... posted. What I found was that soil organisms (worms, etc.) can’t get through the fabric to draw leaves and other organic matter down into the soil. So they go elsewhere where there is no fabric. Their tunnels in the soil aid air circulation and water movement. As the worms etc. go elsewhere, those tunnels collapse and the soil ends up dry, hard, with considerably reduced organic matter! That might be OK for a path, but is not good for a planting bed! Landscape fabric is OK to keep gravel and sand in the base layer under paving from migrating too quickly into the soil, but it is never good in a planting bed IME.

I have not used cardboard but have used paper to help in establishing beds - in my case I was turning sod upside down, covering it with wet newspaper, adding a layer of chopped leaves on top, and then leaving it to rot down over the winter, then planted into it in the spring. I was mainly trying to discourage the grass roots from growing by blocking light, and giving worms time and conditions to aid the decomposition. That worked great for me and was the method I used to start most of the backyard beds here. So the paper was gone in 6-8 months and the grass, paper, and leaves had been converted to useful soil for spring planting.