On the definition of pseudoentropy
Yes, we can always say that $X$ has pseudoentropy at least $H(X)$. You can take $Y$ to be a completely separate, independent random variable that has the same distribution as $X$. Then $X$ and $Y$ are computationally indistinguishable (indeed they are "completely" indistinguishable): $\Pr[A(X) = 1] = \Pr[A(Y) = 1]$ for all algorithms $A$.

Instead of thinking of psuedoentropy as a property of a random variable, maybe it is more helpful to think of it as a property of a distribution. Not a big difference but emphasizes that, for instance, $X$ and $Y$ are never correlated in the definition.