When did the use of paratroopers become common place? Are there any notable paratrooper battles or attacks? Please include countries besides the United States like France, Russia, Israel, Korea, or China. The reason I ask is I read briefly about of the Nazi paratrooper attack at Fort Eben-Emael[1] in The Rise and Fall of the Third Reich by William L. Shirer.
WWII saw the first extensive use of paratroopers. Earlier attempts were stymied by limitations on transport aircraft that couldn't haul large numbers of soldiers that were so heavily laden as a paratrooper. You actually cite one of the first dramatic uses of such troops, at Eben-Emael. The fort should have held out a long while against a conventional ground assault, but the vertical approaches were lightly defended and overwhelmed.
The next use of a combat jump is in the Battle of Crete, beginning in May of 1941. This was a German assault on the island, defended by Greeks, Brits, and from halfway around the world some Aussies and Kiwis. While the attack was successful, losses were crippling. As such, German paratroopers were not sent into combat from the skies for the rest of the war. Like their Italian counterparts, they were used as elite ground infantry.
The invasions of Sicily and Normandy saw US and British paratroopers used on a large scale. Neither invasion went to plan. Friendly fire sent many Allied paratroopers into the sea on Sicily, and the urge to avoid anti-aircraft fire scattered the paratroopers in Normandy. The former was a tragedy, the latter was a lucky happenstance. While many paratroopers died in swamps and the like in Normandy, their scattered nature made it impossible for the Germans to effectively counter-attack them. Operation Market Garden also employed paratroopers to a large degree, with the British troops taking huge losses at Arnhem while waiting at the end of a "bridge too far". I won't include more as you asked for the US to be minimized.
One particular example of non-American paratroops would be the French deployment of Airborne elements of the Foreign Legion to hopefully relieve the siege of Dien Bien Phu by the Viet Mihn during the French Indochina war. It stands to date as the lowest (in altitude of drop) deployment of paratroops in military history. The relief effort did not succeed and Dien Bien Phu fell. It is rumored that the Legionnaires are buried where they fell, upright and facing West, forever saluting France.