How did Premarin get its name?

Usually Jim or Jimmy is short for James

Luekemia was named by John H. Benett in 1845 based on it's effects on the human body.

"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...

It is the name of its genus.

binominal nomenclature, binary nomenclature, or binomial classification system. so yes