It is, but it is UNUSUAL. The natural word would be "bring" or "get." It is almost automatic to add the word "please" for politeness: "Please bring me a glass of water." "Please get me a glass of water." "Bring me a glass of water, please." "Get me a glass of water, please." Without the word "please" it is a command or an order. With the word "please" it's a request. In the United States a diner would use the word "please" even to a waitperson in a restaurant, and a boss would use the word "please" to a subordinate.

Yes it is correct, but it would not be a very polite way of saying it. "Fetch" is a word used when dogs retrieve sticks. (The owner throws a stick and the dog "fetches" it). So it is not necessarily a word you'd want to use when directed at another person — unless with friends in a very informal, joking sort of way. OR could be something a King or Queen would say to his/her servants. Along those lines :)

Yes, especially in the American South.