I don't know if a German doctor could have been issued an Erma pistol, but American doctors and nurses were issued Colt Model 1903/1908 pocket pistols. Medical personnel were allowed to be armed for personal defense and defense of their patients, but only with "defensive" weapons; the standard M1911A1 was considered an "offensive" weapon. Medical personnel could refuse a gun if they wished, and I understand many did.
As to German WWII calibers, the 9mm Kurz (.380 ACP) was not in the German military or police supply system. Pistols in that caliber captured as a result of German occupation of countries like Czechoslovakia and Belgium were seized and used, but new manufacture was in the standard 7.65mm. Of course, the Germans made 9mm Kurz and used captured ammo in that caliber, but it was not a standard caliber.