All those weapons can safely fire any .30-'06 issue ammunition. That does not mean that users of all weapons were issued all ammo. Belt-fed machineguns normally were issued ammo in belts loaded 4 AP (or ball) and 1 tracer. M1 rifle clips were normally loaded with AP. Ball was used in training but rarely was issued in combat because of the need to penetrate light armor, vehicles and the like. Specialty ammo, such as incendiary and frangible ball, was issued in limited quantities for special needs.
BAR ammo was usually AP, issued in boxes, with magazines loaded by the gunner and assistant gunner. M1 clips were occasionally loaded from boxes, but as a rule they were pre-loaded, issued in cans containing bandoliers of six clips each. Ammunition for the M1903 was issued in bandoliers containing 60 rounds of 5 round clips, normally also AP.
.30 Carbine ammunition could be used only in the .30 Carbine; it is not the same as other .30 caliber ammo.
FWIW, from late 1941 to late 1945, Frankford Arsenal, the army's main ammunition factory, produced about 1.5 million rounds of .30 ammunition a day. Contractor factories produced at about the same rate, but started later.
Jim