How did the Hague get its name?

If you mean the city of 'The Hague', it is called: 'Den Haag'.

'The Hague' is called 'Den Haag' in the Netherlands.

Usually Jim or Jimmy is short for James

Tunis

"your name is" is correct English and when used in this way means both your full name or a part of your name (usually surname, depending on the context). "your names are" is also correct English, but...