Are discrete random variables, with same domain and uniform probability, always independent? If $X$ and $Y$ are discrete random variables, both with domain $\{-1,0,1\}$, with uniform probability, does this imply they are independent? What would be the expected value $\mathbb{E}(XY)$ and probability like $\mathbb{P}(XY=1)$ or $\mathbb{P}(XY=-1)$ in this case?

It is simple to construct an example where both variables are marginally uniformly distributed, but they are not independent.  The simplest example is to take $X \sim \text{U} \{ -1,0,1 \}$ and let $Y=X$.  In this case both of the variables have a uniform distribution, but they are perfectly correlated.
