Is the union of all elliptic curves $\mathbb{R}^2$? An elliptic curve
could be written as
$$y^2 = x^3 + a x + b \;.$$

Q1. Is it the case that every point $p \in \mathbb{R}^2$ lies
  on some elliptic curve?
Q2. And what is the natural generalization of this question to higher
  dimensions $d$, $p \in \mathbb{R}^d$?

I ask these questions (obviously!) naively. Thanks for enlightening me!

Not sure about the second question, but here is my take on the first.
Let $p = (p_1, p_2)$. 
Suppose $p_2^2 \neq p_1^3$, then $p$ lies on $y^2 = x^3 + (p_2^2 - p_1^3)$; note, here $\Delta = -432(p_2^2 - p_1^3)^2 \neq 0$, so $y^2 = x^3 + (p_2^2 - p_1^3)$ is non-singular.
If $p_2^2 = p_1^3$, then $p$ lies on $y^2 = x^3 + p_1x - p_1^2$, but $\Delta = -16(4p_1^3 + 27p_1^4) = -16p_1^3(4+27p_1)$ which is non-zero unless $p_1 = 0$ or $p_1 = -\frac{4}{27}$. However, as $p_1^3 = p_2^2$, $p_1^3 \geq 0$ so $p_1 \geq 0$ so $p_1 \neq -\frac{4}{27}$.
If $p_2^2 = p_1^3$ and $p_1 = 0$, then $p = (0, 0)$. As $p$ lies on $y^2 = x^3 + ax + b$, $b = 0$; note that $a$ can be chosen arbitrarily. For the curve $y^2 = x^3 + ax$, $\Delta = -64a^3$, so as long as $a \neq 0$, the curve is non-singular, so $y^2 = x^3 + x$ will do.
In summary, for every $p = (p_1, p_2) \in \mathbb{R}^2$ there is an elliptic curve $L_p$ such that $p \in L_p$. In particular, one choice of $L_p$ is
$$L_p = \begin{cases}
y^2 = x^3 + (p_2^2 - p_1^3) & \text{if}\ p_2^2 \neq p_1^3\\
y^2 = x^3 + p_1x - p_1^2 & \text{if}\ p_2^2 = p_1^3, p \neq (0, 0)\\
y^2 = x^3 + x & \text{if}\ p = (0, 0).
\end{cases}$$
