Spooning out an exquisitely ripe avocado is great if you're going to be making guacamole or any other salsa or sauce with them. If they're a hair less than ripe, you put cut side down on a plate and carefully lift the skin, eventually inverting it, leaving the whole avocado flesh intact on the plate. That's for the "fan" of avocado topping a soup or entree.
The better way to cut a mango is to cut down the side of the pit, hold the nearly half mango in your hand, skin side down, and make a criss cross of knife cuts that don't cut the skin. Then invert the skin and either chow down on perfect little squares of mango or "shave" them into a bowl, ready to be combined with other ingredients. Scraping it uncut into a glass like it was shown in this article works but you've still got a very messy half mango to cut into usable form.
The onion one was spot on. With the cut side of the root end down on the cutting board, onion tears are held to a minimum as you cut the onion into very regular pieces in no time at all.