Yes, stomping still exists in small batches. Clean and preferably covered feet are a must as any fungus or bacteria could ruin the wine..
No, this is a myth
Yes, and the fungus from you feet do make the wine taste "unique"
While personal style is part of the answer. The real reason is for aeration of each type of wine. Certain wines need a wider mouth to breathe, while others flavors are maintained by not allowing to much air. Champagne for example would go flat faster in a wide mouth glass.
Wine is supposed to be drank straight out of the box
This is done solely for business reasons. More styles, more sales.
Who cares, where does the wine end up?
They would take them to their home country
Nowhere, you don't bury survivors.