That is false. There was a prototype called the Persuader constructed in 1918. But, it was nowhere near ready for mass production, much less actual combat use. In regards to US use of the Thompson in WWII, it was used in every theater there were US troops present. It was the standard submachine gun of the US military beginning in 1938, when it was adopted as the M1928A1.
The M2 Carbine saw no appreciable use during the war as production began too late for any guns to reach troops before the end of hostilities.
That is incorrect. The M3 was in use during WWII. The M3A1 began production in 1945 with 15,469 manufactured before the contract was canceled due to the end of the war. It went on to be used in the Korean and later conflicts, with a further 33,000 manufactured by the Ithaca Gun Company