"Wall-to-wall carpets in every room" is a singular noun phrase in the sentence Wall-to-wall carpets in every room is their dream. The phrase represents a single dream (meaning a desire, a hope for the future) so that's why the verb is "is".
A simpler sentence with the same grammatical structure is:
A trip to Europe is my dream.
What if the sentence was:
Wall-to-wall carpets in every room is expensive.
Now, one might see a blur between a verbless clause (where having is implied) and a noun phrase as the subject-verb disagreement is more clear. However, carpets can mean or be replaced with carpeting so the phrase still strongly suggests a usual noun phrase.
Here is a more complex situation:
Are fast cars in cities really reasonable?
It sounds right, right? The question form tends to have a subject-verb agreement.
Is fast cars in cities really reasonable? [might sound off without using "having"]
and the answer can be:
[Having] fast cars in cities is really reasonable.
Because of this situation, there is an additional category of verbless clauses called nominal verbless clause, which appears to be coined by Quirk (a British linguist) but mentioned as debatable. It can also be described as a noun phrase having the elements of a verbless clause. The version of "A Comprehensive Grammar of the English Language" by Quirk et al. (1985) I've found on Google Books has this excerpt:
The nominal verbless clause is a more debatable category than the other nominal clauses. The category seems to be required to account for construtions which, although superficially noun phrases, have some of the semantic and structural characteristics of clauses:
A friend in need is a friend indeed. [proverb]
Wall-to-wall carpets in every room is their dream.