Learn Hardcore Icelandic: Leiðbeinandinn sendir okkur hljóðbók eftir hverja kennslustund. - The instructor sends us an audiobook after each lesson.

So grammatically it’s:
leiðbeinandinn (subject, nominative)
sendir (verb)
okkur (indirect object, dative)
hljóðbók (direct object, accusative)

Why is it hljóðbók and not hljóðbókin?

Hljóðbók means an audiobook (indefinite), while hljóðbókin would mean the audiobook (definite).

The Icelandic sentence talks about what the tutor sends each time in a general way, not about one specific known audiobook, so the indefinite form fits better:

sendir okkur hljóðbók = sends us an audiobook
sendir okkur hljóðbókina = sends us the audiobook (some specific one already known in the context)

Grammatically, hljóðbók here is in the accusative singular as the direct object.

What does eftir mean here, and why does eftir hverja kennslustund use the accusative?

In this context, eftir means after (in time).

Eftir can be followed by different cases depending on meaning, but in time expressions like after each lesson, the noun phrase is usually in the accusative.

hverja = feminine accusative singular of hver (each/every)
kennslustund = lesson/class, here also accusative singular

So eftir hverja kennslustund literally is after each lesson, with kennslustund (a feminine noun) in the accusative, and hverja agreeing with it.

Why is it hverja and not just hver in eftir hverja kennslustund?

Hver is a pronoun/adjective that declines for gender, number, and case to agree with the noun it describes.

Here it modifies kennslustund, which is:

gender: feminine
number: singular
case: accusative (because of eftir in this time expression)

So we use the feminine accusative singular form:

masculine acc. sg.: hvern
neuter acc. sg.: hvert
feminine acc. sg.: hverja

Therefore: eftir hverja kennslustund = after each lesson.

What exactly is kennslustund, and how is it formed?

Kennslustund is a compound noun:

kennsla = teaching / instruction
stund = hour / period / time

Put together, kennslustund literally means something like teaching-time, and in practice just means a class or a lesson.

In the sentence, kennslustund is in the accusative singular, governed by eftir:

nominative: kennslustund
accusative: kennslustund (same form for this word)

Is the word order sendir okkur hljóðbók fixed, or can it be changed?

The most natural neutral order is:

Verb – indirect object – direct object
→ sendir okkur hljóðbók

You can sometimes move elements around for emphasis, but not all orders sound equally natural. For example:

Leiðbeinandinn sendir okkur hljóðbók… (neutral)
Leiðbeinandinn sendir hljóðbók til okkar… (grammatical, but til okkar changes the feel; it’s more like “to us” as a prepositional phrase, and is less usual here than plain dative okkur)

For a learner, it’s best to treat [verb] + [dative pronoun] + [accusative noun] as the default pattern for verbs like senda (send), gefa (give), sýna (show), etc.

What is the difference between leiðbeinandi and kennari?

Both can be translated as teacher in some contexts, but they are not identical:

kennari = teacher in the more formal or standard sense (school teacher, university lecturer, etc.)
leiðbeinandi = instructor, tutor, supervisor, guide; often someone who guides you more individually (e.g. a thesis supervisor, a tutor, a coach, a guide in a course)

So leiðbeinandinn suggests more of a tutor/instructor role rather than a regular classroom teacher.

Does the present tense sendir mean “sends”, “will send”, or “usually sends”?

Icelandic present tense is quite flexible and can cover several English uses:

In this sentence, sendir is best understood as habitual present:

Leiðbeinandinn sendir okkur hljóðbók eftir hverja kennslustund.
→ The tutor sends us an audiobook after each lesson (every time, as a rule).

Depending on context, Icelandic present can correspond to:

simple present: he sends
present progressive: he is sending
or even a near-future meaning: he is going to send / he will send

Here, eftir hverja kennslustund clearly makes it a repeated, habitual action.

How do you pronounce leiðbeinandinn and hljóðbók approximately?

Very roughly, in an English-friendly approximation:

leiðbeinandinn
- leið ≈ “layth” (with a soft voiced th, as in this)
- beina ≈ “bay-na”
- ndinn ≈ “n-din” (short i, like din)
- Whole: something like “LAYTH-bay-nan-din” (with Icelandic vowels; this is just approximate)
hljóðbók
- hlj-: the h is pronounced before the l, a bit like blowing out air before saying lyo
- óð: roughly like English “oath”, but with a long ó and a soft ð (as in this)
- bók: like “boke” with a long o
- Whole: roughly “HLYO-th-boke” (again, this is an approximation)

For actual learning, listening to native audio is important, especially for sounds like ð, hlj-, and long vowels.