I live in Hawaii, on Maui, along the Hana Highway in Nahiku, a rainforest. I have bananas growing in my jungle yard plot parcel space here. I make banana bread everyday, almost, and I peel about 3 pounds of banana a day, and I know exactly that dark, toward black "base" of the banana that you mention as the icky part.
I can not tell you exactly what it is called, but I can tell you that it is even ickier on the outside when still attached to the stalk. Small stiff black material is at the terminal end of each fruit and I peel them off, and get a drop of sticky sap, to keep the bananas nicer. As the flower peels layers of petals off, each reveals a hand of bananas, and the tiny fruit have a yellow hairlike projection then, which I believe to be the external and internal part you remove before eating. According to the Wikipedia page on seedless fruit ( the banana ), the fertilization process is aborted, so it is my suspicion that the black spike terminal end of the seedless fruit is indeed a no longer neccesary remnant of the flower, perhaps the most distal end of the pistil??
But the rest of the flower, which looks like the whole of the flower, is hanging at the end of the stalk. It peels itself off, petal after petal, and then some of those tiny fingers turn into bananas and the rest fall to the ground like droppings from bats, fingers, sap and petals in a pile. The tight bundle of petals at the end is purple and good to eat, albeit quite stringent. A honeymooning couple stoped and showed me how to prepare it one afternoon.
As to the second question about how people peel bananas, I peel too many bananas to be going the length, I peel the banana around, like removing a wrap-around skirt, and then bend at the end in the final gesture to remove the terminal spike. It actually pulls out, leaving a small indentation, and sometimes even revealing the three natural divisions of the fruit.