What does Amadeus mean?
A Latin name combining amare 'to love' (amadeo 'I love') with Deus 'God', understood as 'love of God' or 'loving God'. The am- root is unambiguous; best known from Wolfgang Amadeus Mozart.
What does Regis mean?
From Latin 'regis', the genitive of 'rex' — literally 'of the king', i.e. royal/kingly. Popularized partly via Saint Jean-Francois Regis. Means 'of/belonging to the king' rather than 'king' itself.