You're correct, if the orientational vector is unitary. Otherwise you must calculate the unitary vector in the direction of $\vec{O}_1$ and then perform the projection (just divide by the norm of the orientational vector). The perpendicular component is just the rejection vector of $\Delta \vec{R}$, given by the definition: $\Delta\vec{R}_{\perp} = \Delta\vec{R} - \Delta\vec{R}_{\parallel} $.
If you draw the vectors you will be able to visualize that this is correct and then convince yourself. Take a look:
You can determine straight forward the rotation by the internal product between the two orientational vectors $\vec{O}_1$ and $\vec{O}_2$.
$\theta = \arccos{\left( \frac{\vec{O}_1 \cdot \vec{O}_2}{|\vec{O}_1||\vec{O}_2|}\right)} $