Learn Hardcore Icelandic: Ég þurrka hendurnar með handklæðinu eftir sturtu. - I dry my hands with the towel after a shower.

hendurnar = the hands (definite, plural). Icelandic often uses the definite form for body parts in natural sentences like this, similar to English I wash my hands (but Icelandic tends to choose definiteness differently).
hendur would be hands (indefinite plural), which can sound less natural here unless you mean some hands in a more general sense.
You can add possession, but it’s usually unnecessary:
- Ég þurrka hendurnar. = I dry (my) hands.
- Ég þurrka hendurnar mínar. = I dry my hands (more emphatic/contrastive).

What case is hendurnar, and why that case?

Hendurnar is accusative plural definite of hönd (hand).
It’s accusative because þurrka is a transitive verb here: you’re drying something (the hands), so they are the direct object.

(Quick table, singular/plural):

nominative: hönd / hendur
accusative: hönd / hendur
dative: hendi / höndum
genitive: handar / handa

Definite plural forms include -nar: hendurnar.

Why does með cause handklæðinu to change form?

Because the preposition með normally takes the dative case (when it means with in the sense of using/accompanied by).
So:

nominative: handklæði
accusative: handklæði
dative: handklæði
genitive: handklæðis

And with the definite article in dative singular:

handklæðinu = the towel (dative singular definite)

Why is it handklæðinu (singular, definite) and not plural or indefinite?

handklæðinu implies one specific towel (often the one you’re using right now).
If you mean towels in general, you could say:
- Ég þurrka hendurnar með handklæði. = I dry my hands with a towel.
If multiple towels are involved:
- ... með handklæðum = ... with towels (dative plural)

Why is it eftir sturtu and not eftir sturtuna?

Both are possible, with a nuance:

eftir sturtu = after showering / after a shower (more general/routine)
eftir sturtuna = after the shower (a specific shower you just had, or one already mentioned)

What case is sturtu, and why?

Sturtu is accusative singular of sturta (shower).
The preposition eftir typically takes:

accusative when it means after (time)
dative in some other meanings/uses (more location/“following along”), but for time after, accusative is the normal choice.

So: eftir + accusative → eftir sturtu.

Is the word order fixed? Could I move parts around?

The given word order is very natural:
Ég (subject) + þurrka (verb) + hendurnar (object) + með ... (instrument) + eftir ... (time).

You can move time phrases, especially for emphasis:

Eftir sturtu þurrka ég hendurnar með handklæðinu. (After showering, I dry my hands with the towel.) But in a neutral statement, the original order is the most straightforward.

How do I pronounce Ég, þurrka, hendurnar, and especially þ / ð?

Key sounds:

þ is like English th in thin.
ð is like English th in this (often softer, sometimes almost disappears depending on position).

Rough pronunciation guide:

Ég ≈ yeh-gh (the g is a soft voiced sound; many learners approximate it)
þurrka: starts with th (thin), u is like Icelandic u (not English ur), and rk is a clear cluster
hendurnar: stress on the first syllable: HEN-...
með: like meth but with voiced th (as in this)

Could I use another verb besides þurrka (like “dry off”)?

Yes. Að þurrka is the straightforward to dry (something).
If you want to express drying yourself off, Icelandic often uses reflexive phrasing, for example:

Ég þurrka mér. = I dry myself off.
Ég þurrka mér um hendurnar. = I wipe/dry my hands (idiomatic).

But for the simple idea “dry the hands,” Ég þurrka hendurnar ... is perfectly normal.