Let's start with the dandelions.  "Self-regulate" probably means the plant adjusts to various environments (too much water, too little water, sunlight variations, etc).  Calcium deficiency may benefit dandelions because other competing plants (such as turfgrass) suffer before the dandelions.  However, adding calcium will not control dandelions.  For control of dandelions, read here:
https://extension.umn.edu/weeds/dandelions

Next, the calcium.  While calcium is a necessary soil nutrient, it is not typically the limiting nutrient.  That is, most soils have plenty.  Get your soil tested for P and K (the usual limiting nutrients) and pH (acidity):
https://soiltest.cfans.umn.edu/testing-services/lawn-garden
You can also ask them to test for calcium (an extra fee).

Clay and compacted soils are a challenge for growing anything:
https://www.ndsu.edu/agriculture/extension/publications/evaluating-preparing-and-amending-lawn-and-garden-soil

Read here about plants that do well in clay:
https://www.uvm.edu/d10-files/documents/2024-10/Best_Plants_for_30_Tough_Sites_UMN.pdf

Finally, clover (instead of grass).  There are lots of resources on starting non-turfgrass lawns.  Look here:
https://turf.umn.edu/testing-alternative-clover-options-use-kentucky-bluegrass-lawns
https://naturalresources.extension.iastate.edu/lawn-alternatives
https://extension.illinois.edu/blogs/ilriverhort/2016-03-18-microcloversa-new-lawn-alternative

There are many nurseries that will supply you with seeds (and even "install" the alt-lawn).