Hi, Persona Grata. "Make" is this sense does mean "be" or "become", not "manufacture." "Rabbits make good pets; grizzly bears don't make good pets." You might also hear something like "I think you would make a good doctor."
Literally, "I don't understand" means simply "I don't understand," but "I can't understand" means "I am not capable of understanding. -- it is impossible for me to understand." Often, though, people would say "I can't understand" as a more emphatic version of "I don't understand" -- especially to express annoyance or exasperation. "I can't understand how you can lose you car keys every single morining!!" It indicates that you have tried and tried to understand, but you still don't understand.