Group action by conjugacy and "curves"

Question

Let $G$ be a group, $H\unlhd G$ and "$\cdot$" the $G$-action by conjugacy on $H$:

\begin{alignat*}{1} \cdot:G \times H &\longrightarrow H \\ (g,h) &\longmapsto g \cdot h:= g^{-1}hg\\ \end{alignat*}

Say $g' \stackrel{h}{\sim} g \stackrel{(def.)}{\Longleftrightarrow} g' \cdot h = g \cdot h$. I see that $\stackrel{h}{\sim}$ is an equivalence relation in $G$.

My question is twofold (possibly merging together for something I'm missing):

1. Is there any link between the two partitions of $G$, namely $G/H$ and $\lbrace [g]_{\stackrel{h}{\sim}}, g \in G \rbrace$?
2. Given $\tilde g \in G, h \in H$, call $R_\tilde g(h):=[\tilde g]_{\stackrel{h}{\sim}}$ and $O(h):=\lbrace g^{-1}hg, g \in G\rbrace$; how is $R_\tilde g(h) \cap O(h)$ made? In particular, is it empty?