Yes. The character table of SL(2,q) is known. If q is odd, the sum of the dimensions of the irreducible representations of SL(2,q) is q²+q. If q is even, this sum is q². In either case, this is bigger than the dimension of $\mathbb{C}\{X\}$, so there must exist irreps not in $\mathbb{C}\{X\}$.

Yes. The character table of SL(2,q) is known. If q is odd, the sum of the dimensions of the irreducible representations of SL(2,q) is q²+q. If q is even, this sum is q². In either case, this is bigger than the dimension of $\mathbb{C}\{X\}$, so there must exist irreps not in $\mathbb{C}\{X\}$.