I buy organic potatoes. Does this mean they are like heirloom potatoes and I can grow new potatoes from the spuds sprouting in my closet? Or might they be genetically modified and sterile (or whatever it is when they’re plants)?

Yes, each little eye will grow a new potato. Just cut the potatos in fourths

Great!!
When buying seeds, do they have to be heirloom or heritage in order to be able to save seeds and use them again? Hybrids won't set sets, will they?