For any real number, $r=0$ if and only if $r^2=0$, so "set[ting] $r=0$" is the same as setting $r^2$ to zero. Equivalently: if $r=0$, then $r^2=0$, so of course you get that $r^2=0$.
However, I don't understand why you think you get $\sin(2\theta)=0$. If $r^2=0$, then $-4\sin(\theta)=0$. That means that $\sin(\theta)=0$; where did that $2$ come from?
If you set $\theta=0$ instead, then $\sin(\theta) = \sin(0)$. How much is $\sin(0)$? How much is that when multiplied by $-4$? And what is the (only) value of $r$ that will make $r^2 = -4\sin(0)$ true?
Again, I don't understand why you think you will get "Does not exist" if you plug in $\theta=0$. This is simply not the case. (Though, if you had $r^2 = -4\cos(\theta)$, and tried to find a real value of $r$ for the case $\theta=0$, then you would be unable to find one; are you sure you are computing $\sin(0)$ correctly?)