Let's examine our statements in turn.

• Statement I is true. From the diagram, $m \angle A = 90^\circ$ so $\angle A$ is a right angle.

• Statement II is false. Angles $\angle B$ and $\angle C$ are not vertical since they have a common side.

• Statement III is false. We can see from the diagram that $\angle B < 180^\circ.$ So, $\angle B$ is not a reflex angle.

So, the correct answer is "I only."

Let's examine our statements in turn.

• Statement I is false. From the diagram, $\angle A$ and the angle with measure $60^\circ$ are vertical angles. Therefore, \[m\angle A = 60^\circ < 90^\circ.\] Consequently, $\angle A$ is an acute angle.

• Statement II is true. From the diagram, $\angle C$ and the angle with measure $40^\circ$ are vertical angles. Therefore, \[m\angle C = 40^\circ < 90^\circ.\] Consequently, $\angle C$ is an acute angle.

• Statement III is true. Angles $\angle B$ and $\angle D$ are vertical since they are opposed to one another.

So, the correct answer is "II and III only."

Since $\angle S$ is a straight angle, we have \[ m\angle S=180^\circ. \]

Since $\angle T$ is acute, we have \[ 0^\circ < m\angle T < 90^\circ. \]

So the sum of these two angles is restricted by the following inequality:

\begin{align} 180^\circ + 0^\circ &< m\angle S + m\angle T < 180^\circ + 90^\circ \\[3pt] 180^\circ &< m\angle S + m\angle T < 270^\circ \end{align}

Consequently, the sum of $\angle S$ and $\angle T$ is a reflex angle.

Since $\angle S$ is a null and a right angle, we have \[ m\angle S=0^\circ \] and since $\angle T$ is a right angle, we have \[ m\angle T=90^{\circ}. \]

So, their sum is

\[ m\angle S + m\angle T=0^{\circ} + 90^{\circ}=90^{\circ}. \]

Consequently, the sum of $\angle S$ and $\angle T$ is a right angle.

Since $\angle VOZ$ is a right angle, we have \begin{align*} m \angle VOZ = 90^\circ,\qquad m \angle VOW = m \angle YOZ = 15^\circ. \end{align*}

Therefore, we can form an equation and solve for $t$ as follows:

\begin{align*} m \angle VOW+ m \angle WOX + m \angle XOY + m \angle YOZ &= m \angle VOZ \\[3pt] 15^\circ+t + t + 15^\circ &= 90^\circ \\[3pt] 2t &= 60^\circ \\[3pt] t &= 30^\circ \end{align*}

Note that $\angle AOD$ is a straight angle and $\angle BOC$ is a right angle. Hence, \begin{align*} m \angle AOD = 180^\circ,\qquad m \angle BOC = 90^\circ. \end{align*}

Therefore, we can form an equation and solve for $x$ as follows:

\begin{align*} m \angle AOB + m \angle BOC + m \angle COD &= m \angle AOD \\[3pt] x + 90^\circ + x &= 180^\circ \\[3pt] 2x &= 90^\circ \\[3pt] x &= 45^\circ \end{align*}

From the diagram, $\angle AOD$ is a straight angle, so \[ m \angle AOD = 180^\circ. \]

Now, the angle addition postulate gives:

\begin{align*} m \angle AOB + m \angle BOC + m \angle COD &= m \angle AOD \\[3pt] 80^{\circ} + m \angle BOC + 30^\circ &= 180^\circ \\[3pt] m \angle BOC &= 70^\circ \end{align*}

From the diagram, $\angle AOC$ and $\angle EOG$ are vertical angles. Since vertical angles are congruent, we have

\[m\angle AOC = m\angle EOG = 90^\circ.\]

On the other hand, since $\angle AOC$ is a right angle, the angle addition postulate gives:

\begin{align*} m \angle AOB + m \angle BOC &= m \angle AOC \\[3pt] 30^\circ + m \angle BOC &= 90^\circ \\[3pt] m \angle BOC &= 60^\circ \end{align*}

Then, given that $m \angle BOD=120^{\circ}$, we have:

\begin{align*} m \angle BOC + m \angle COD &= m \angle BOD \\[3pt] 60^{\circ} + m \angle COD &= 120^{\circ} \\[3pt] m \angle COD &= 60^{\circ} \end{align*}

Finally, $\angle COD$ and $\angle GOH$ are vertical angles. Therefore,

\[m\angle GOH=m\angle COD=60^\circ.\]