The number of elements in ${\mathcal O}_{gH}$ is the index of $H\cap gHg^{-1}$ in $H$. Your condition 1 means that there is an element $g$ such that $|H:H\cap gHg^{-1}|\lt |H:H\cap g^{-1}Hg|=|gHg^{-1}: H\cap gHg^{-1}|$. So you want $H\cap gHg^{-1}$ to have different but finite indices in $H$ and in $gHg^{-1}$.
The best way to achieve what you want (including 2 and 3) is to consider a group $H$ with two isomorphic subgroups $U,V$ of different finite indices, then consider the HNN extension $\langle H,g\mid gUg^{-1}=V\rangle$. 

There are many such groups. For example, the Thompson group $F$ has a subgroup of index 2 that is isomorphic to itself (see <a href="http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CDMQFjAA&url=http%3A%2F%2Fwww.maths.nuigalway.ie%2F~chew%2Fpapers%2Fdwnlds%2FcomF.pdf.gz&ei=GJKDT7mhLMe8twfHxvyECA&usg=AFQjCNHt2pZEUsZnQkjt43sod3UTTMnPow">this paper</a>, Corollary 3.3, where all subgroups of finite index in $F$ which are isomorphic to $F$ are described. 

Another source of examples is the lamplighter group $L=\mathbb{Z}_2\wr \mathbb{Z}$. If $a$ is the generators of $\mathbb{Z}_2$ and $b$ is the generator of $\mathbb{Z}$, then $L_k=\langle a, b^k\rangle$ is isomorphic to $L$ and has a finite index in $L$. The isomorphism from $L$ to $L_k$ is $a\mapsto a$, $b\mapsto b^k$ (here $k\ge 1$). 

I think it is worthwhile to check whether properties 2 and 3 holds for these examples. 

<b>Warning</b> I will leave the rest of the previous answer as an illustration of the fact that everybody should remember to use multiplication table properly. Thanks to Ian Agol and  Dave Penneys for pointing out my error.

For example, take $H=F_2\times F_3$, $U=U_1\times U_2$ where $U_1$ is of index 10 in $F_2$, $U_2$ is of index 12 in $F_3$ (in that case, by Schreier's formula, the rank of $U_1$ is 11, the rank of $U_2$ is 25), $V=V_1\times V_2$ with $V_1$ of index 24, $V_2$ of index $5$ (then the rank of $V_1$ is 25 and the rank of $V_2$ is 11). In that case $U$ is isomorphic to $V$. The other properties should follow from the general properties of HNN extensions.