Learn Hardcore Icelandic: Þær drekka kaffi á morgnana.

The letter Þ/þ is called thorn and is pronounced like the th in thing, never like the th in this.
Þær is pronounced roughly like thair (as in English there, but with the th sound from thing at the start, and with a trilled or tapped r at the end).
The vowel æ is a diphthong, pronounced like eye in English.

Why is the verb drekka used here, and how is it conjugated?

Drekka is the verb to drink. In the present tense it conjugates (simplified) as:

ég drekk – I drink
þú drekkur – you (sg.) drink
hann / hún / það drekkur – he / she / it drinks
við drekkum – we drink
þið drekkið – you (pl.) drink
þeir / þær / þau drekka – they drink

Because the subject is Þær (they, feminine plural), the correct form is drekka.

Why doesn’t Icelandic say something like “are drinking”? Why just drekka?

Icelandic does not have a separate -ing form like English continuous tenses (are drinking, are going).
The simple present drekka can mean both they drink (habitually) and they are drinking (right now), depending on context.
If you really want to emphasize an ongoing action, you can use a construction like Þær eru að drekka kaffi, but it is not needed in this sentence.

Why is there no word for a or the before kaffi?

Icelandic has no indefinite article like English a/an. So kaffi on its own can mean coffee or (some) coffee.
The definite article the is usually attached as an ending, not written as a separate word.
So kaffi = coffee, while kaffið = the coffee.

What is the difference between kaffi and kaffið?

Both refer to coffee, but kaffið is definite: the coffee, some specific coffee already known from context.
In this sentence, we are talking about coffee in general – what they usually drink in the mornings – so kaffi is used.
Saying Þær drekka kaffið á morgnana would sound more like They drink the coffee in the mornings (for example, a particular pot of coffee you both know about).

What does á morgnana literally mean, and why that form morgnana?

Literally, á morgnana is on the mornings, but idiomatically it means in the mornings (habitually).
Morgunn is the noun morning. Its accusative plural definite form is morgnana (from the paradigm: morgunn – morgnar – morgna – morgnana).
So the structure is: á (on/in) + morgnana (the mornings) → on the mornings / in the mornings.

Why does Icelandic use á morgnana for in the mornings, and not some other preposition?

The preposition á is very flexible and can mean on, in, at, to, depending on context.
With time expressions, á often means on / in (during), as in:

á daginn – in the daytime
á kvöldin – in the evenings
á veturna – in the winters

Á takes the accusative in á morgnana, so morgnana is accusative plural definite.

What is the difference between á morgnana, á morgun, and í morgun?

á morgnana – in the mornings, regularly/habitually (every morning or most mornings).
á morgun – tomorrow (this is a fixed expression; here morgun is an adverb meaning tomorrow).
í morgun – this morning (earlier today in the morning).

So á morgnana describes a repeated habit, not a single specific morning.

Why is morgnana in that particular case and number?

Because of á and the meaning during the mornings, morgnana is in the accusative plural definite.
Plural, because we are talking about mornings in general, not just one morning; definite, because Icelandic often uses a definite plural form in habitual time expressions.
Even though the Icelandic is formally definite (the mornings), English normally translates this kind of phrase as just mornings.

Why does kaffi not change its form as an object? Is there a case ending missing?

Kaffi is a neuter noun whose nominative, accusative, and dative singular forms all look the same: kaffi.
In this sentence, kaffi is the direct object of drekka, so it is in the accusative singular, but that happens to be identical in form to the nominative.
You only see a different form in the genitive singular (kaffis) or in the plural.

Is the word order fixed? Can I move á morgnana, or turn this into a question?

The basic declarative order here is Subject – Verb – Object – Time: Þær drekka kaffi á morgnana.
Because Icelandic is a verb‑second (V2) language, you can put á morgnana at the front, but then the verb must still be in second position:
Á morgnana drekka þær kaffi – In the mornings, they drink coffee.
To make a yes/no question, you typically put the verb first:
Drekka þær kaffi á morgnana? – Do they drink coffee in the mornings?