Show: "Checking no solution for system of linear equations with integer variables and coefficients" $\in \mathbf{NP}$
There is a polynomial time algorithm that determines whether a system of linear equations over the integers has a solution. The algorithm uses Hermite normal form, which can be computed in polynomial time. See lecture notes of Swastik Kopparty: Hermite normal form and finding integer solutions.

Your proof that your problem is in coNP is incomplete, since you haven't shown that the witness has polynomial size. Assuming you did manage to show it, all this implies is that the problem is in coNP, not that it is coNP-hard. There's absolutely no issue for a problem to be in both NP and coNP. Indeed, every problem in P is also in NP and coNP. If you can put a problem in both NP and coNP, then this suggests that it might actually be in P, though it is conjectured that $\mathsf{P} \subsetneq \mathsf{NP} \cap \mathsf{coNP}$. One problem which might lie in the middle is (a decision version of) integer factorization.