He is a duke, and a prince, as well as a few other things. But no royal ever goes by a name like 'Prince Duke Earl Charles". Instead, the person picks one title to be known by. Most people will...
Because he was invested with the title by Queen Elizabeth II.
the day he was born would be my guess.
He is the Prince of Wales because he is the next in line to be King of the United Kingdom. "Prince of Wales" is one of his titles.
The fox usually gets shot to make sure it is dead, then it's tail is cut off and used as a "trophy".