Elizabeth nor Charles are claiming devine right. They are a unfiying vestige of times past, providing as she did a human constant, an embodiment of the Commonwealth.
So long as their heredity isn't overtly providing them the ability or write or enforce law, it does not seem an affront to democracy.
While Elizabeth has CHOSEN to not use these powers much, any future monarch can. Just look at the U.S. in the last 6 years to see what happens when a country relies more on historical norms rather than law.
Not really. In practice these powers belong to parliament and the monarch performs a merely ceremonial role; actions are performed in his or her name but not at her behest. In the English constitution parliament is sovereign and the monarch acts on its instructions.
She held the power of royal prerogative but couldn't ever exercise it because Parliament retains the right to dismiss and choose a new monarch anytime they like.
The issue of royal prerogative was settled in the Glorious Revolution when Parliament decided it didn't like the King, James II and just selected a new one.
Every year we remind the monarch at the State Opening of Parliament that they can't ever use their royal prerogative.
The monarch might have influence but ultimate power rests with Parliament.