If $X$ and $Y$ are separable metric spaces, then the Borel $\sigma$-algebra $B(X \times Y)$ of the product is the $\sigma$-algebra generated by $B(X)\times B(Y)$. I am embarrassed to admit that I don't know the answers to:

Question 1. What is a counterexample when $X$ and $Y$ are non-separable?

Question 2. If $X$ is an uncountable discrete metric space, does $B(X) \times B(X)$ generate the Borel $\sigma$-algebra on $X \times X$?

Question 3. If $X$ and $Y$ are metric spaces, with $X$ separable, does $B(X) \times B(Y)$ generate the Borel $\sigma$-algebra on $X \times Y$?