Why are some x-coordinates unsuitable for an ECDSA generator point? For Bitcoin's ECDSA curve (secp256k1, where $a=0$, $b=7$), why can't the generator point's first coordinate be $x=0$?
That is, the point on the curve would be $(0,y)$ where $y$ satisfies $y^2 = 0^3 + 7=7$.

Are there any elements other than $0$ which can't be used as a generator point's $x$ coordinate?
Here is a plot of secp256k1 over the reals. That I'll use for illustration purposes.



What happens at $x=-3$, well, we get $y^2 = (-3)^3+7$ which reduces to $y^2=-20$. Solving for $y$ we get $y=\sqrt{-20}$, which is not in the reals, so, we cannot have a point on that curve for $x=-3$. 

This is what is happening. In the case of the real secp256k1 used in bitcoin, we are using a finite field instead of the reals. In the reals, the equation $y^2=x^3+7$ is not defined for the entire domain. Similarly, over the finite field, the equation is not defined over the entire field.

In other words, in order for there to be a point (actually 2 points) on the curve at $x$, we must be able to take the square root of $x^3+7$ in the field. As to the number of valid $x$ values, see this question on Math.SE.