We could take any $Z$ of mean 1 (excluding constants) and any independent X. $Y$ could be defined conditionally as$$Y \vert Z,X \sim \mathcal{D}\Big(f(Z), \quad aZ+bX^2-f(Z)^2\Big),$$where $\mathcal{D}(\mu, \sigma^2)$ is a generic distribution of mean $\mu$ and variance $\sigma^2$, and $f$ is any function such that$$ E\left(f(Z)\right)=0.$$Now 1,2,3, and 5 come by construction, and using$$ E\left(Y^2\right) = \mathrm{Var}(Y) + E\left(Y\right)^2 $$you should get 4.
We only need to prove that the variance can always be non-negative.
If we have no restrictions on the support of $X$ and $Z$, one could easily make sure that$$bX^2 > f(Z)^2-aZ$$for $f(Z) = Z-1$.
