A typical process passes ambient air over some liquid or solid solvent that can absorb CO2, then later inserts energy to separate the CO2 again for storage. For example, sodium hydroxide reacts with CO2 in the air to form sodium carbonate. Then later, the sodium carbonate is heated to release pure CO2, regenerating the sodium hydroxide in the process.
This doesn't violate any laws of thermodynamics because of the constant energy required. Compared to growing trees, direct air capture is generally much more expensive, requiring large capital investment and constant energy input. It is more space efficient though.
Ah yeah, no absolutely not. In total it takes much more energy to capture the CO2 than was generated by burning the fossil fuel that emitted it.
I'm not too familiar with algea/moss CO2 absorption, but it could be better. Usually the downside of a vertical structure is you increase the capital investment again, negating the advantage of plants. And to provide lighting you'll need energy which takes space as well (e.g. solar panel field)