To determine if $X$ receives a coin of denomination $5$, we need to analyze the possible combinations of three coins from the denominations $1$, $2$, $5$, $10$, and $20$.

Let's denote the coins as $a$, $b$, and $c$ where $a < b < c$ and $a, b, c \in \{1, 2, 5, 10, 20\}$. The possible combinations of three coins are:

1. $1, 2, 5$
2. $1, 2, 10$
3. $1, 2, 20$
4. $1, 5, 10$
5. $1, 5, 20$
6. $1, 10, 20$
7. $2, 5, 10$
8. $2, 5, 20$
9. $2, 10, 20$
10. $5, 10, 20$

Now, let's evaluate the statements:

Statement I: $m$ is not a prime number.

• If $m$ is not a prime number, it could be any of the sums from the combinations above that are not prime. However, this does not directly help us determine if a coin of denomination $5$ is included, as both combinations with and without $5$ can result in non-prime sums.

Statement II: The sum of the digits of $m$ is greater than $5$.

• This statement alone does not help us determine if a coin of denomination $5$ is included, as both combinations with and without $5$ can have sums whose digits add up to more than $5$.

Using both statements together:

• We need to find a combination where the sum is not a prime number and the sum of its digits is greater than $5$.
• Consider the combination $1, 2, 10$: The sum is $13$, which is a prime number, so it does not satisfy Statement I.
• Consider the combination $1, 5, 10$: The sum is $16$, which is not a prime number, and the sum of its digits is $1 + 6 = 7$, which is greater than $5$. This satisfies both statements.
• Consider the combination $1, 2, 20$: The sum is $23$, which is a prime number, so it does not satisfy Statement I.
• Consider the combination $2, 5, 10$: The sum is $17$, which is a prime number, so it does not satisfy Statement I.

Thus, the only combination that satisfies both statements is $1, 5, 10$, which includes a coin of denomination $5$.

Therefore, the question can be answered using both statements together, but cannot be answered using either statement alone. The correct option is C.