Can submarines shoot down aircraft

Not anymore. In both world wars submarines were equipped with anti-aircraft weapons for defending themselves while surfaced, but they made the subs slower while submerged, and enemy aircraft usually stayed out of their range while calling in a destroyer by radio. After World War 2 submarines became streamlined to improve their speed, and their AA guns were removed. The prevailing school of thought is that a submarine has a better chance of survival underwater than trying to fight on the surface. The British Navy experimented with a missile launched from submarines to shoot down aircraft, but even still the aircraft holds most of the advantages, and the submarine is probably more apt to survive by hiding under the waves.