Carbon dioxide in lime ground
Limestone is the natural sedimentary rock, e.g., the top of Mt. Everest that began as sea floor.  "Storing" $\ce{CO2}$ in limestone requires heating calcium carbonate white hot to dissociate it to lime, then scrubbing "other" $\ce{CO2}$ with the lime.  This is, to coin an adjective, smartless, since you also burned fuel to heat the limestone. Curing concrete imbibes and traps $\ce{CO2}$, but it was initially dissociated to release it.

Your second equation is terrible. You obtain a strong base by adding acid to a salt.  $\ce{CaCO3}$ plus carbonic acid, $\ce{CO2}$ in water, makes water-soluble calcium bicarbonate $\ce{Ca(HCO3)2}$.  That is how limestone caves are dissolved out, and why karst topography is stark (surface erosion) and unstable (sinkholes).

To vastly store carbon at a profit, plant forests, harvest, and build wood-frame buildings.  The photosynthesis bottleneck, carbon fixation, depends upon the slowest least efficient enzyme on Earth, RuBisCO.  A smart (at least, not corrupt) society then

1) Gene-gineers a better RuBisCO.  Ten times the productivity leaves it the worst enzyme on Earth.  Everybody would have a full belly.

2) Gene-gineers a 10x fast-growing forest with huge lignin content in its wood.  It won't rot, it won't feed termites, it will be immensely strong.  If wood-frame buildings become unfashionable, bury the wood in deserts to store carbon.

There is no problem so easily solved that social advocacy will not make it intractable, disastrous, and expensive.  The unabled maximize self-esteem to be credible - effectively selecting for the deluded, hence advocacy.  A society inundated with rules from religious and secular political classes ignoring productive ends will collapse.  USSR to Pol Pot, reality is what it appears to be - thermodynamics.  There is no cheating possible.