The answer is C

(2) tells us that x and y are both positive
If we apply this result to (1), we see that both x and y are positive AND y must be greater than x in order for x/y < /1/2

If x<y, then x-y will always be a negative number, and negative numbers are always less than 1 (the original question asks "is x-y >1" and the answer is no)

Sure thing.

If y were negative, then we wouldn't be able to find sqrt(y).
For the purposes of the GMAT, we can't find the sqrt of a negative number. So, in this case, we know that y must be positive.

Hmm, now that I have written my explanation, I don't like my original question. As it is currently written, it leaves room for some ambiguity.
While the GMAT tests only our knowledge of real numbers, the test writers are often careful to avoid contradicting other areas of math. At an early age, we learn that we can't find the square root of a negative number. Then, as we progress through mathematics, we later learn about the realm of complex (imaginary) numbers, where the sqrt(-1) = i and i^2=-1 and when we allow sqrt(y) to be a compex number then y^2 can be a negative number.

My current question allows for sqrt(y) to be an imaginary number, in which case y (and y^2) can be positive or negative. In which case the answer is E.

Had I added the proviso "x and y are real numbers" then the answer would be C, since we sqrt(negative number) is undefined and we can't take an undefined number and square it and get some real number.

I hope that helps.