If you insist on capturing CO₂ then capture it somewhere it is available in abundance instead of ~400 parts per million: at the source. Capture it at gas turbine exhausts, at coal-burning power station smoke stacks, maybe on board ships if the power is available to do so. Use some other process for capturing atmospheric CO₂, e.g. lime stone capture. Trying to actively capture airborne CO₂ is like creating needle-in-haystack-finding factories, it is possible but a waste of resources.
No, that would just postpone whatever problem existed in the first place since you'd be left with a huge mass of compressed exhaust gases which do not have any use. Since those exhaust gases consist mostly of nitrogen (due to it forming ~80% of the atmosphere) this would also entail a massive waste of energy expended in compressing that nitrogen.
Separating out the CO₂ would make it possible to either use it in some form where it replaces CO₂ otherwise produced by other means - e.g. for welding, cooling, carbonation of liquids etc. - or to sequester it in some way.
