1. .303 was standard calibre; the calculations about weight of fire leading to the decision that 8 guns were needed were based around this, IIRC.
2. Once .303 was proven to be a bit lacking (late 1940), ISTR reading that the plan was for fighters to have 20mm, with bombers getting .5. After Pearl Harbor, the Americans needed all the .5 they could get, which substantially delayed fitting weapons of this calibre to RAF aircraft.
My memory may well be flawed on both accounts, apart from the fact that .303 was the standard calibre (and appeared adequate at the time the aircraft were designed).