Is $\mathbb{R}$ a subset of $\mathbb{R} \times \mathbb{R}$? So I'm curious as why $\mathbb{R} \subseteq \mathbb{R}^{2}$, since $\mathbb{R}^{2} = \mathbb{R} \times \mathbb{R} = \left \{ (a,b) \mid a\in \mathbb{R}, b\in \mathbb{R} \right \}$. Do we think of $\mathbb{R}$ as being $\mathbb{R} \times 0$ in this case?  

Technically no, for the reason you stated. However there IS a canonical bijection between $\Bbb{R}$ and $\Bbb{R}\times \{0\}$.
