Let $G$ be a reductive group acting on an affine variety $X$. For simplicity, one may assume $G=SL_n$ or $G=U_n$ and assume the field is $\mathbb C$. Given this one can show $\mathbb C[X]^G$ is finitely generated algebra.

Question:(1) Each of the orbit closures $\overline{Gx}$ is a finite union of $G$-orbits.

(2) Each of the orbit closures $\overline{Gx}$ contains a unique closed orbit which has minimal dimension among these $G$-orbits.

I guess the two questions might be related to a result saying that $$ \pi(x_1)=\pi(x_2) \ \ \ \ \Longleftrightarrow \ \ \ \ \overline{G x_1} \cap \overline{G x_2} \neq \varnothing $$ where $\pi: X\to X //G\equiv Spec \mathbb C[X]^G$ is the GIT quotient.

EDIT: For question (2), suppose $Gx$ is not closed. If $y\in \overline{Gx}$ but $y\not \in Gx$, then what relation can we expect between $\overline{Gx}$ and $\overline{Gy}$? For example, do we have $\dim\overline{Gy} < \dim\overline{Gx}$, or $\dim G_y \ge 1$?