The size of the knife comes down to personal preference rather than the size of the onion.
Small “kitchen knives” are mostly used in the home. The more you become attached to “your” knife, the sooner it becomes a universal tool. Experienced chefs also use their large knife to peel onions.
The bottom part of the bulb is left intact in almost all cases.
Cut off a cap at the top of the onion, you can now see the ring arrangement of the individual layers.
You cut along the onion from the top to the base into the skin or the first layer and can now remove the onion skins.