Learn Hardcore Icelandic: Smjörið er kalt í dag.

Smjörið is smjör (butter) + the suffixed definite article -ið (neuter singular).
So smjörið means the butter (some specific butter you’re talking about, e.g., the butter on the counter).

How do I know smjör is neuter, and why does that matter in this sentence?

Smjör is a neuter noun in Icelandic. Gender matters because adjectives agree with the noun’s gender/number/case.
Since the subject is neuter singular, the adjective appears as kalt (neuter singular), not kaldur (masculine) or köld (feminine).

Why is the adjective kalt and not something like kaldið?

Kalt is an adjective in its indefinite form used predicatively after vera (to be). Icelandic normally does not add the definite article to adjectives in this kind of simple “X is Y” statement.
So: Smjörið er kalt = The butter is cold.

What are the other forms of kaldur (cold), and how would the sentence change with a different noun?

Basic nominative singular forms are:

Masculine: kaldur
Feminine: köld
Neuter: kalt

Examples:

Kaffið er kalt. (kaffi = neuter)
Súpan er köld. (súpa = feminine)
Bjórinn er kaldur. (bjór = masculine)

What grammatical case is smjörið in here?

It’s nominative, because it’s the subject of the sentence.
The verb er (from vera) doesn’t force an object case here; it links the subject to a description (kalt).

Is er just the present tense of vera, and does it change with person/number?

Yes—er is the present tense form meaning is. The present tense conjugation is:

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)

So Smjörið er kalt uses 3rd person singular (it is).

What does í dag literally mean, and why is it í?

Í dag literally means in (the) day, used idiomatically as today.
Icelandic commonly uses í with time expressions like this: í dag, í gær (yesterday is different wording), í kvöld (tonight), etc.

What case is dag in í dag?

It’s accusative singular (dag is the accusative form of dagur, “day”).
With í, Icelandic often uses:

accusative for time/duration or motion into something
dative for location/static position

Í dag is a fixed time expression and uses the accusative form dag.

Could I drop the article and say Smjör er kalt í dag?

You can say smjör without the article, but it changes the feel:

Smjörið er kalt í dag. = The butter (this butter) is cold today.
Smjör er kalt. would sound more like a general statement (Butter is cold), which is unusual as a general truth.

In everyday speech, if you mean the butter you’re dealing with, smjörið is the natural choice.

Where does í dag usually go in the sentence? Can it move?

The most neutral placement is at the end: Smjörið er kalt í dag.
You can move it for emphasis, but Icelandic word order has rules (especially about the verb being early in main clauses). A common alternative is to front the time expression for emphasis:

Í dag er smjörið kalt. = Today, the butter is cold.

How is Smjörið er kalt í dag pronounced, especially the ð?

A careful, learner-friendly guide:

Smjörið: roughly SMYÖ-riþ (the ö is like the vowel in German schön; ð is like the th in this)
er: like ehr
kalt: like kalt with a clear t
í: like ee
dag: roughly daag, ending with a soft throaty sound for g in many accents

Also, Icelandic stress is usually on the first syllable: SMJÖ-rið, KALT.