I think the question is very broad and cannot find precise mechanisms. I am not sure about the precise mechanisms of the regulation.
I think they are at least Vitamin D, diffusion and specific calcium channels: calbindins. More specifically, calbindin-D9K and the basolateral calcium pump transport calcium from cell interior to extracellular space.
The calcium gradient should also be free, -2-6 mmol/l, in the lumen to allow diffusional calcium absorption from the interstitial space. Similarly, free calcium concentration in the extracellular fluid of the lamina propria.
I think that the most important mechanism for this absorption is the electrochemical gradient of sodium across the epithelial cell boundary of the lumen, similarly as for carbohydrates. To keep the extracellular space free from calcium, the NA/K-ATPase i.e. sodium pump has to work.
I did not find any hormones that affect the intestinal calcium absorption. Which factors affect calcium absorption and its regulation?