How to obtain calcium carbonate?
Most of the $\ce{CaCO3}$ in commercial use today is mined not synthesized. The one exception I know a bit about is the ultra-small particle sized materials. However, few of those are pure, they have surface treatments which constitute a significant portion of their total weight. Pharmaceutical and perhaps food grades are also synthesized.

The major synthetic pathway is circular: limestone is mined, then it is heated which produces lime and carbon dioxide. The lime is then mixed with water and $\ce{CO2}$ is added (bubbled in/through) and $\ce{CaCO3}$ precipitates. It is then dried.

Purification depends on the source; how impure it is to start with. It is generally difficult to separate magnesium carbonate from calcium carbonate, and given that limestone and aragonite as well as dolomite ($\ce{CaMg(CO3)2}$) are taken out of the ground (i.e. are 'dirt' cheap) there's usually no need.

As far as the risks of making it, well you could burn yourself, adding lime too fast to water could splash and it's infamous for dissolving bodies in murder mysteries, as well as being used in real life to help in outhouses and gardens where organic material is desired to rapidly decompose. Meaning you don't want to get it in your eyes, nor do you want it left on your skin for very long. It can, theoretically, fume, while being decomposed and that wouldn't be good to breathe. And if a centrifuge or mechanical press is used to squeeze the water out of it (first step in drying), then you have the assorted mechanical dangers. Its synthesis is probably one of the safest chemical syntheses you could undertake.

$\ce{CaCO3}$ decomposes with fairly mild heating (I think around 840 °C), so any material which is predominantly that (like seashells, bones) could be used to make the lime.