Actually the British monarch has a lot of power, de jure, they just don't use it due to "tradition". They're the ones who appoint prime ministers, and they're the ones who dissolve parliament for new elections. Traditionally they do those things at the behest of others, but de jure it's their right. (Note: those two powers are the ones abused by Hindenburg in Weimar Germany to de facto appoint whomever as chancellor based on his power to dissolve the Bundestag if they disagreed with his choice)
No. Royal prerogative is only exercised on advice of the prime minister, the cabinet or by the consent of parliament. Constitutional convention is not just tradition. And the fact that the constitution of the UK is uncodified doesn't mean the monarch has any de jure powers. Even the prorogation of parliament on advice of the prime minister was held to be unlawful: https://en.wikipedia.org/wiki/R_(Miller)_v_The_Prime_Ministe...
