For a set of 3 numbers, assuming there is only one mode,

I think that above solution is correct and the answer is A, though there is one case missing.

Set can be of a form:
A. {X,Y,Y}
B. {X,X,Y}
OR
C. {XXX}

Basically telling us that there is only one mode, stem is saying that we do not have three distinct numbers in the set.

(1) The median equals the range:

A. {X,Y,Y} --> Y=Y-X --> X=0 --> Set: {0,Y,Y}. Mode=Y, Range=Y-0=Y, --> Y=Y. True.
B. {X,X,Y} --> X=Y-X --> 2X=Y --> Set: {X,X,2X}. Mode=X, Range=2X-X=X --> X=X. True.
C. {XXX} --> X=X-X --> X=0 --> Set: {0,0,0}. Mode=0, Range=0 --> 0=0. True.

Sufficient.

(2) The largest number is twice the value of the smallest number:

A. {X,Y,Y} --> Y=2X --> Set: {X,2X,2X}. Mode=2X, Range=2X-X=X, --> 2X equals to X only if X=0 (set: {0,0,0}), but we don't know that. Not always true. For example we can have set: {1,2,2} Mode=2, but range=1, 2#1.
B. {X,X,Y} --> Y=2X --> Set: {X,X,2X}. Mode=X, Range=2X-X=X --> X=X. True.
C. {XXX} --> X=2X --> X=0 --> Set: {0,0,0}. Mode=0, Range=0 --> 0=0. True.

Two different answers (cases B and C always true, A not always).

Not sufficient.

Answer: A.