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?

No, suppose you have $X$ as you described and for $Y$ you flip a coin and in 0.5 probability it takes the value of $X$ otherwise you uniformly sample from $\{-1,0,1\}$.
The marginal probabilities will be uniform yet they are obviously dependent. Short simulation below.
x    <- sample(c(-1, 0, 1), 100000, TRUE)
coin <- rbinom(100000, 1, 0.5)
y    <- coin * sample(c(-1, 0, 1), 100000, TRUE) + (1 - coin) * x

     
> cor(x, y)
[1] 0.4986911


You can see that $X$ any $Y$ are dependent and have the same marginal uniform distribution.
