Translation:She is an important woman.
Do we use the -e at the end of the adjective because it's "de vrouw"? What if it were a het noun... such as "een moeilijke paard"?
That's right, the -e at the end of adjectives is used for de-words and plurals, or if the noun comes with a definite article ("de" or "het"). Since "paard" is a het-word, it would be:
"een moeilijk paard"
See here for more on Dutch adjectives.
belangrijke looks like a word consisting of other words, is it one?
Belang is importance and rijk is rich.
So does it mean important and rich woman? Or just important?
I think it means "rich in importance" - so just important, not rich money-wise.
What is the difference between woman and lady. I'm just curious.