After the time shown, when will the two hands FIRST point in exactly opposite directions?

At 7:00 the minute hand is $210^{\circ }$ behind the hour hand. Opposite means $180^{\circ }$ behind, so it must gain only $30^{\circ }:\frac{30}{5.5}=5\frac{5}{11}$ min past 7. (5 min ignores the hour hand's motion.)