Learn Hardcore Icelandic: Nóttin er löng og köld.

Yes. Icelandic usually marks definiteness by adding a suffix to the noun. -in is the nominative singular feminine definite ending, so Nóttin means the night. Without it, nótt means “a night” or just “night” (indefinite). A few other common forms:

Accusative singular definite: nóttina
Dative singular definite: nóttunni (e.g., á nóttunni = “at night”)
Genitive singular definite: næturinnar (genitive sg is nætur: e.g., til nætur = “tonight”)

Is there ever a separate word for “the” before a noun?

In everyday Icelandic, definiteness is almost always shown by the suffix (e.g., nóttin). There is a separate word, hinn/hin/hitt, but it’s formal/literary. You might see something like hin langa nótt (“the long night”) in formal writing, but in normal speech you’d say langa nóttin.

What’s the dictionary form and gender of "nótt"? How do I make the plural?

The dictionary form is nótt (feminine). The plural nominative is nætur (“nights”). Example: Nætur eru langar og kaldar (“Nights are long and cold”). The word is irregular: note ó → æ in plural and in the genitive singular (nætur).

What does "er" mean here, and how is it conjugated?

er is “is,” the 3rd person singular present of vera (“to be”). Present tense:

ég er (I am)
þú ert (you are)
hann/hún/það er (he/she/it is)
við erum (we are)
þið eruð (you pl. are)
þeir/þær/þau eru (they are)

Why are the adjectives "löng" and "köld" shaped that way?

Adjectives must agree with the noun in gender, number, and case. Nóttin is feminine singular nominative, so the adjectives are feminine singular nominative. The base forms are:

long: langur (m), löng (f), langt (n)
cold: kaldur (m), köld (f), kalt (n) Hence: Nóttin er löng og köld.

Why not “langa” and “kalda” instead?

After “to be” (in predicative position), adjectives normally take the strong forms, which here are löng and köld. The weak forms langa/kalda are used when the adjective stands before a definite noun (attributive):

langa og kalda nóttin = “the long, cold night” (attributive, definite)
But: Nóttin er löng og köld (predicative)

How do I say “The nights are long and cold”?

Use the feminine plural. Definite: Næturnar eru langar og kaldar. (Indefinite: Nætur eru langar og kaldar.)

What would change if the subject were a different gender?

Adjectives would match that gender:

Masculine: Dagurinn er langur og kaldur (“The day is long and cold”).
Neuter: Kvöldið er langt og kalt (“The evening is long and cold”).

Can I drop the article for a general statement, like “Night is long and cold”?

You could say Nótt er löng og köld, but it feels odd in Icelandic. For general truths, plural is common: Nætur eru langar og kaldar (“Nights are long and cold”). Using the definite singular (Nóttin) also works when context makes it generic (e.g., talking about a season or place).

How do I negate this sentence?

Place ekki after the verb: Nóttin er ekki löng og köld (“The night is not long and cold”). For “neither … nor,” use hvorki … né: Nóttin er hvorki löng né köld.

How is "og" pronounced, and what does it mean?

og means “and.” It’s often pronounced simply like [o]/[ɔ], with the g very soft or silent in casual speech.

Any quick pronunciation tips for the whole sentence?

Nóttin: ó is like a pure “o” (no diphthong), and tt has a slight “h-ish” breath before it (preaspiration), roughly like “NOHT-in.”
er: like “ehr.”
löng: ö like French “eu” in “peur”; ng sounds like “ngk” → “leuhngk.”
og: usually just “o.”
köld: ö as above; ld sounds like “lt” → “kœlt.” Stress the first syllable of each word: Ícelandic always stresses the first syllable.