While looking for answers to these questions, one looks back through the history of the doughnut to find how it was described in early cookbooks. In some of the earliest cookbooks, the doughnut is described as a cruller or fried cakes. These fried cakes were usually rolled into twists or were little balls, such as the olykoeks of the Dutch influence. Other cookbooks were sometimes even vaguer. They would direct the baker to “make the cakes in the size of a half dollar” or “roll out and shape the way you see fit”. None of these descriptions ever gave the shape we know and love today.
The closest doughnut shape of the modern doughnut was the diamond shaped doughnuts described in Directions for Cookery, in its Various Branches cookbook, by Eliza Leslie. Within she describes how to prepare a doughnut, from the amount of yeast to flour ratio, how long to let rise with the yeast, up to the point where you fry the dough. Before you fry the dough, Eliza tells you to “cut it into thick diamond-shaped cakes with a jagging iron. If you find the dough so soft as to be unmanageable, mix in a little more flour; but not else. Have ready a skillet of boiling lard; put the doughnuts into it, and fry them brown; and when cool grate loaf-sugar over them.” No mention as to why the diamond shape was cut into the dough, but one can assume it was to make the frying that much easier.
Ultimately, reason doughnuts have holes in them because of the way they are cooked; doughnuts are fried. Since they are so dense and full inside, the heat used during cooking wouldn't be able to reach all the way into the middle to cook the center if these yummy baked goods were simply disc-shaped with no hole. The rapidness of the cooking procedure caused the dough to not fully cook. Now, this was combated earlier by both the Dutch and Germans by placing fruit, nuts, or something in the center that did not need to be cooked, but later, the hole was placed there to give more surface area for the doughnut to fully cook all the way around. This may have come due to trial and error, as no one can say for sure when the first hole happened.
The most interesting of the story of the hole originate with a gentleman called Captain Hanson Gregory. Known as the “inventor of the doughnut hole”, Gregory would tell the story of how he invented the hole. (Two stories that is) The first story entailed that he liked to munch on fried cakes while steering his craft, hence being a Captain. One day, in 1847, the seas were rough and he needed both hands to control the rudder. So, he slapped several cakes on the spoke of his wheel, making holes. As an old man, he liked to tell his other story of when he was a boy he had been watching his mother frying doughnuts and had noticed that the centers always remained partially uncooked and doughy. 'Mother', he said, "leave a hole in the center." Laughingly, she obliged him and never went back to the old way. Her method was widely copied. Another story from Gregory was that he did not like the nuts in the center of the doughnuts, so he would poke them out, and the result left a hole.
All three stories are fun filled, and Gregory to this day is still known as the inventor of the doughnut hole.