Lateral roots break through the cortex on their way out but they arise from deeper within the root. The pericycle is just inside the endodermis and remains meristematic. Right, if they formed ahead of this zone they would be torn off as the tip advances through the soil. Once they are in the pericycle, water and solutes can take any path to the xylem. Yes most of the root mass is cortex which is made up of parenchyma cells and provides a site of storage of starch and organic nitrogen compounds.
Internodes have less meristematic potential than other parts of the stem.
Nodes retain meristematic potential longer than internodes and this is where new roots as well as new shoots tend to grow out. A carrot is indeed, although not everything that grows in the ground turns out to be a root.