For every $x$ the equation $x=yaya^{-1}$ has a solution. Indeed, replace $ya$ by $z$. We get $x=z^2a^{-2}$ or $xa^2=z^2$. But in $SL_n(\mathbb{C})$ every element is a square, so $z$ can be found, hence $y=za^{-1}$ exists. 

<b> Update 1. </b> In order to see that every complex non-singular matrix is a square, it is enough to consider a Jordan block $J_n(a)$ ($a$'s on the diagonal, 1 on the next diagonal, the rest are 0). If we denote $b=\sqrt{a}$, then a square root of $J_n(a)$ is a triangular matrix, it has $b$ on the diagonal,
$2/b$ on the next diagonal, $-\frac{1}{8b^3}$ on the second diagonal, $\frac{1}{16b^5}$ on the third diagonal, etc. The number on the diagonal \# $i$ is equal to $c_i/(b^{2i-1})$ where $c_i$ can be defined by induction, it does not depend on $a$. Also if you represent your matrix as $A=\exp(B)$ for some $B$, then the square root of $A$ is just $\exp(B/2)$. Note that we need $a\ne 0$, so this does not work for singular matrices.  

<b> Update 2. </b> About the equation $y=x(x^{-1})^t$. Note that this equation is stable under conjugation by complex orthogonal matrices (i.e. matrices $a$ with $a^t=a^{-1}$).Thus instead of $y$ we can consider  $aya^{-1}$ with $a$ orthogonal. Hence we can assume that $y$ is triangular. For $n=2$ this immediately gives:

** A triangular matrix $y\in SL_2(\mathbb{C})$ is of the form $x(x^{-1})^t$ if and only if either $y=1$ or the eigenvales of $y$ are not equal to 1. Thus $y$ is of that form iff either $y=1$ or $y$ is not unipotent. 

<b> Update 3 </b> For $n=3$ the description is more complicated. For example, all uni-upper triangular representable matrices $A$ have the form
$$\left(\begin{array}{lll} 1 & ca &cb \\\ 0 & 1 & c\\\  0& 0 & 1\end{array}\right)$$
Hence if $A[3,2]=0$, $A$ must be equal to 1. On the other hand if $c\ne 0$, $A$ is arbitrary.