For $x\in G$, define $H_x=\{g^{-1}xg \mid g\in G\}$. Under what conditions on $x$ will $H_x \leq G$. Further, if $H_x \leq G$, will $H_x \lhd G$? For $x\in G$, define $H_x=\{g^{-1}xg \mid g\in G\}$. Under what conditions on $x$ will $H_x \leq G$. Further, if $H_x \leq G$, will $H_x \lhd G$?
My attempt is as below: 
$G$ is a group so $e\in G$ and $e^{-1}xe=x \in H_x$.This means that $H_x$ will be non-empty. 
For $H_x$ to be a subgroup we must show that for any $g_1^{-1}x g_1$ and $g_2^{-1}xg_2$ in $H_x$ , $(g_1^{-1}xg_1)(g_2^{-1}xg_2)^{-1}\in H_x$.
Also $(g_1^{-1}xg_1)(g_2^{-1}xg_2)^{-1}=(g_1^{-1}xg_1g_2^{-1}x^{-1}g_2)$. 
Now if $x=e$ then $x \in H_x \Rightarrow e\in H_x$ and  we have $(g_1^{-1}xg_1g_2^{-1}x^{-1}g_2)=e \in H_x$. So We can say that $H_x \leq G$. In this case since $H_x=\{e\}$, we have $H_x \lhd G$.
Please let me know if my reasoning is right. Also is $x=e$ the only possible condition on $x$ which makes $H_x$ a subgroup?

Your reasoning is correct. To answer your final question, think that something to hold for every $g_1, g_2$ must hold in particular for $g_1=g_2$. Explicitly:
\begin{alignat}{1}
H_x \le G &\iff \forall g_1,g_2 \in G, \exists g\in G \mid g_1^{-1}xg_1(g_2^{-1}xg_2)^{-1}=g^{-1}xg \\
&\Longrightarrow \forall g_1 \in G, \exists g\in G \mid g_1^{-1}xg_1(g_1^{-1}xg_1)^{-1}=e=g^{-1}xg \\
&\Longrightarrow \exists g\in G \mid xg=g \\
&\Longrightarrow x=e
\end{alignat}
You have already shown that $x=e \Longrightarrow H_x \le G$, so indeed:  $H_x \le G \iff x=e$.
