If x & y are independent are they uncorrelated?

Answer 1

if they are independent they could be correlated or might not be.
think of a deck of cards
let your first draw be x and 2nd draw be y
each draw is independent (meaning the first draw wont determine what you will get on the second draw) (just cuz u draw a clubs on ur first draw doesnt mean your second draw has to be a club too)

Answer 2

If they are truly independent, they are not correlated. There is "wiggle room" in this, so you have to stick to definitions.