Let $X = V + W$ and $Y = V + Z$ where $V, W, Z$ are independent Pois($\lambda$) random variables.
I have been able to show that Cov(X,Y) = Var(V) = $\lambda$ which implies that X and Y are not independent.
How would I determine if X and Y are conditionally independent given V.
What is the calculation for P(X = 5, Y = 0) in terms of λ
For your first question, check if $$P(X=x, Y=y \mid V=v) = P(X=x \mid V=v) P(Y=y \mid V=v)$$ holds for $x \ge v$ and $y \ge v$.
For your second question, use the law of total probability to obtain $$P(X=5, Y=0) = \sum_{v = 0}^\infty P(X=5, Y=0, V=v).$$ Note that the terms on the right are zero if $v > 5$.
In both steps above, it will be helpful to write things in terms of $V, W,Z$ rather than $V,X,Y$.