German Fallschirmjagers jumped without their guns and had to retrieve them from drop containers in order to fight.

By contrast both US and British paratroopers jumped with their rifles.  We had cases and they didn't?

What about the Soviets?  BTW, I know some of them were dropped into the snow from 75 ft.  LANO (low altitude, no opening b/c no chute).  Soviets figure the snow was deep enough for a safe landing.  It worked.  But they were ass up in the air and the Finns shot them that way.

NVM: guess the Rooskies carried carbines.

They did jump with what guns they could - about a quarter jumped with MP40s, but the standard weapon for FJs was the same as the standard weapon for standard infantry - the Kar98, but this and their MG34s were in the weapon canisters dropped with them (the Kar98 was too big to safely get through the jump door with it). This worked just fine for early jumps into Norway, France, Belgium, etc - those landings were pretty much uncontested, a few extra minutes to grab gear was not a problem.
Their parachute was hot garbage - a terrible single riser job that required you land face-forward. You see pictures of the FJs with big thick kneepads so they will hopefully not shatter their kneecaps on landing. Also, the opening of that parachute was allegedly so violent that it sometimes broke bones. A wonderful piece of equipment for sure. Looking at it, I'm wondering if the off axis weight of a leg bag would have you spinning like a top all the way down. Suppressive fire by vomit is a thing, I guess.

In Crete (1941) they ran into 4x the number of troops they were told they would run into, and those troops knew they were coming - so they got shot up badly. They still managed to win - but it was costly enough Hitler seems to have forbidden all major airborne operations after that. There were some smaller ones (which mostly seem to have used gliders - like the Mussolini rescue), but never anything near this scale again, FJs became elite infantry for the most part. You do see the FG42 come out for the FJs after Crete, likely in answer to the situation experienced there - a full power, 20rd capacity semi auto rifle with full auto capability that was short enough to jump with.

US and UK knew what happened on Crete, and could incorporate those lessons learned into their own airborne programs, avoiding some of the pitfalls. They also had much better parachutes to begin with, and planes that were safer to jump out of with bulky gear (late war anyway - I think early war Brit paratrooper delivery aircraft were also rather terrifying).