"Uniformly distributed" is slightly counter-intuitive here, but seems to be intended to mean that the probabilities for each possible value are either $0$ or some constant $k$ for each possible value in the support, no matter what the pattern of the values are.
(B) is uniformly distributed because $f(S)\to \mathbb R:f(x)=x^3$$f(S)\to \mathbb R:f_B(x)=x^3$ is an injective function so, since $|S|=21$, each possible value in the support of $X^3$ here has probability $\frac1{21}$.
(D) is uniformly distributed because $f(S)\to \mathbb R:f(x)=(x+10)^2$$f(S)\to \mathbb R:f_D(x)=(x+10)^2$ is also an injective function for this particular $S$ so each possible value in the support of $X^3$ again has probability $\frac1{21}$.
(A) and (C) are not uniformly distributed. For example $\mathbb P(X^2=0)=\frac1{21}$ while $\mathbb P(X^2=1)=\frac2{21}$ and similarly $\mathbb P((X-5)^2=0)=\frac1{21}$ while $\mathbb P((X-5)^2=1)=\frac2{21}$.
You do not need an injective functionYour final part of your question is correct: if $\operatorname{rem}(a, b)$ is the remainder when dividing $a$ by $b$, i.e. the amount in $[0,b)$ by which $a$ exceedsyou start with a multiple of $b$,uniformly distributed discrete distribution and then here $\operatorname{rem}(X, 3)$ would be supported on $\{0,1,2\}$ each with probabilityapply an injective function, you get another $\frac13$ and so would be uniformly distributeduniformly distributed discrete distribution. But
You do not always need an injective function: if $\operatorname{rem}(a, b)$ is the remainder when dividing $a$ by $b$, i.e. the amount in $[0,b)$ by which $a$ exceeds a multiple of $b$, then here $\operatorname{rem}(X, 3)$ would be supported on $\{0,1,2\}$ each with probability $\frac13$ and so would be uniformly distributed.
An injective function is insufficient for a uniformly distributed continuous distribution. If you had started with $X\sim U[-10,10]$ uniformly distributed on the real interval, then $X^3$ would not be uniformly distributed despite $f(x)=x^3$ being an injective function.