Is it true, that $\mathbb{R} \times \mathbb{Q} \sim \mathbb{R}$? Just like in title, is it true that $\mathbb{R} \times \mathbb{Q} \sim \mathbb{R}$?
My answer would be yes, since $\mathbb{R}^2 \sim \mathbb{R}$.

If by $\sim$ you mean "there is a bijection between", then the answer is yes, and your argument works fine through Cantor Bernstein's theorem : there is an obvious injection $\mathbb{Q}\to \mathbb{R}$ and so a sequence of injections as follows :
$\mathbb{R}\to \mathbb{R}\times \mathbb{Q} \to \mathbb{R}\times \mathbb{R} \to \mathbb{R}$.  
By Cantor-Bernstein, all these injections "can become" (I'll let you make that precise) bijections
