"Museums are important cultural buildings."

Translation: A múzeumok fontos kulturális épületek.

[BaloghTams19]

I would rather say "institutions" (intézmények) instead of "buildings".

[gr8fuld34d]

Why are "fontos" and "kulturalis" in singular form when they come after a plural noun?

[vvsey]

They are not after a plural noun. They are in front of a noun: "épületek". They do not belong to the subject, they are related to (or part of) the predicate.
And adjectives before nouns stay singular.

[CliveWilto]

Why it is wrong to include vannak?
A múzeumok fontos és kulturális épületek vannak

[Judit294350]

Because you always drop van-vannak in this type of sentence.

[SandraBern295980]

Somebody please can explain why Duolingo accepts Bírók és úgyvédek..., but appoints that is is not correct using Múzeumok without the A before?

[Judit294350]

It depends on the sense of the sentence. Here it is a general statement about museums. But if the other sentence was about some judges and lawyers it is not a general statement and does not have a definite article.