Learn Hardcore Icelandic: Herbergið er kalt.

Icelandic marks definiteness with a suffix rather than a separate article. For neuter nouns in the nominative singular, the definite ending is -ið. So herbergi (“room”) + -ið gives herbergið (“the room”).

What gender is herbergi, and how can I recognize it?

Herbergi is a neuter noun. A common clue is the -i ending in the base form, and its definite form ends in -ið (characteristic of neuter nominative singular). In the plural you’ll also see neuter patterns: indefinite plural herbergi, definite plural herbergin.

Why is the adjective kalt and not kaldur or köld?

Adjectives in Icelandic agree in gender, number, and case with their noun. The masculine base is kaldur (“cold”), the feminine nominative is köld, and the neuter nominative singular is kalt. Since herbergið is neuter singular, we use kalt.

Is kalt here an attributive or predicative adjective, and does that affect its form?

Here kalt is predicative (it follows the verb er = “is”). Predicative adjectives always take the strong inflection, matching gender/number/case (neuter nom. sg. → kalt). Attributive adjectives in front of a definite noun would take the weak form (e.g. kalda in kallaða bókina “the cold book”).

Why is the word order Herbergið er kalt (Subject–Verb–Predicate)? Can I switch it?

The neutral descriptive order in Icelandic is Subject–Verb–Predicate, just like English. You can emphasize something by fronting an adverb or phrase, but with a simple statement you don’t normally invert word order. To form a question, however, you invert Verb and Subject: Er herbergið kalt?

How do I ask “Is the room cold?” in Icelandic?

Invert the verb and subject: Er herbergið kalt? (“Is the room cold?”). Note that the rest of the sentence remains the same.

How would I say “The rooms are cold”?

Use the definite plural herbergin, the plural form of vera, and the neuter plural adjective:
Herbergin eru köld.
Here eru is “are” (3rd pl. of vera) and köld is the strong neuter nominative plural of kaldur.