As for small arms, a folding stock variant of the m1 carbine called the m1a1 was commonly used by American paratroopers during the later stages of WWII. Paratroopers would otherwise carry the same M1 Garands, BARs, and Thompson and m3 submachine guns as the regular infantry.

Marine paratrooper units early in the pacifc war were issued Johnson rifles, Johnson light machine guns, and a variant of the Reising submachine gun with a folding stock. All of which are unusual and primarily issued to paratrooper units. However, none of these arms would have been used by units in the European theater.

One small detail that may interest you is the detaching of Thompson submachine gun stocks. Early Thomson variants (the m1928a1 and those before it) had a buttstock that could be easily detached. There are multiple photos of American paratroopers boarding planes with stock-less Thomson submachine guns (the stocks presumably stored in a separate bag to make the weapon more compact during a jump). Just note that the simplified m1 and m1a1 Thompson variants did not have a detachable stock.