Cubiculum is second declension neuter, which means that it ends with "-um" in the nominative. "Primum" agrees with "cubiculum" so it is also neuter. On the other hand, "lectum" is masculine, however it is in the accusative as it is the object of the sentence, so it ends with "um"
Does the Latin really imply what this English translation does, namely that the other bedrooms do not contain beds? Because I would've gone for "The first bedroom has a bed", except it was one of those choose-the-bubbles exercises, and there was no "a" provided. How does article-less Latin deal with this?
While the general meaning is close, the declensions used for each term do not "allow" this translation.
While we can not know if "primum cubiculum" is in nominative or accusative case, "lectum" is in accusative case, so it cannot be a subject.
Besides, the verb used is "habet" (has) and not "est" (is).
It does not say "a" bed or "the" bed since there are no articles in Latin. Depending on the preceding sentences, it it possible to translate this sentence as:
The first bedroom has a bed The first bedroom has the bed The first bedroom has his bed The first first bedroom had her bed
The choice of article is up to the translater: whichever makes most sense. So both "a bad" or "the bed" should be accepted here.