# Let the king ... the king's official

"You should ... your official" 

# provinces

A province is a large area into which some countries are divided for the purposes of government. See how you translated this in Esther 1:1.

# harem

where the wives of a man with many wives live

# the fortress

This refers to a castle, stronghold or fortified city. See how you translated this in Esther 1:2.

# Susa

See how you translated the name of this place in Esther 1:2

# Let them be put under the care of Hegai, the king's official, who is in charge of the women

"Let Hegai, the king's official, who is in charge of the women, take care of them"

# Hegai

This is a man's name. 

# their cosmetics

A "cosmetic" is a substance such as a cream, lotion, or powder that women usually put on their face or body to improve their appearance.