Learn Hardcore Icelandic: Hann þvær sér í sturtu eftir vinnu. - He washes himself in the shower after work.

sér is the dative form of the reflexive pronoun (3rd person) and means “himself/herself/itself/themselves” depending on the subject.
The verb þvo governs the dative in the set phrase þvo sér “to wash oneself,” so you must use the dative reflexive.
sig is the accusative reflexive; it’s not used here because this verb phrase calls for the dative.
hann would mean “he washes him (some other man),” not himself.

Examples:

Why is the verb form þvær and not þvo?

The vowel change (þvo → þvær) is an i-umlaut type alternation in the present singular.

What case is used in í sturtu, and why?

í takes the dative for location (“in”) and the accusative for motion (“into”).
í sturtu is dative singular of sturta (“shower”) because it denotes location: he is washing while in the shower.
If you mean motion into the shower, use accusative: Hann fer í sturtuna (“He goes into the shower”).

Why is it eftir vinnu and not something with a definite article?

eftir here means “after (in time)” and governs the accusative. vinna (work) has the form vinnu in both accusative and dative singular, so you don’t see a shape difference.
The expression eftir vinnu is idiomatic and generic, like English “after work” (no article).
If you refer to a specific shift or job, you can say eftir vinnuna (“after the [specific] work/shift”).

Could the sentence mean he’s doing it right now?

As written, Hann þvær sér… most naturally reads as a habitual/general statement (“He washes himself …”).
For an ongoing action right now, use the progressive construction: Hann er að þvo sér í sturtu (núna).

Why isn’t it í sturtunni (“in the shower” with the definite ending)?

Icelandic often uses an indefinite noun for routine activities and generic locations: í skóla (at school), í vinnu (at work), í sturtu (in the shower).
Use the definite form when a specific, previously mentioned shower is meant: í sturtunni (“in the [particular] shower”).

Can I move the time phrase to the front? What happens to word order?

Yes. Icelandic is verb-second (V2) in main clauses. If you front the time phrase, the finite verb stays in second position:

How would I say “He washes his hands/hair after work”?

Use the idiom þvo sér um + definite body part:

How do I negate this?

Place ekki after the finite verb:

Hann þvær sér ekki í sturtu eftir vinnu.
If you front something (V2 still applies), ekki still follows the finite verb:
Eftir vinnu þvær hann sér ekki í sturtu.

How do I say it with other persons?

1st sg: Ég þvæ mér (í sturtu eftir vinnu).
2nd sg: Þú þværð þér…
3rd sg: Hann/Hún/Það þvær sér…
3rd pl: Þeir/Þær/Þau þvo sér…
Note: The reflexive forms (sig/sér/sín) are only 3rd person. In 1st/2nd person you use ordinary personal pronouns in the right case (e.g., mér, þér, okkur, ykkur).

Could I drop í sturtu?

Yes. Hann þvær sér eftir vinnu simply says he washes himself after work (could be at a sink, etc.). í sturtu specifies where.

What if he’s washing someone else?

Then you use a normal (non-reflexive) object in the accusative:

Are there natural alternatives to say “take a shower”?

Very common: Hann fer í sturtu (eftir vinnu). = He takes a shower.
Hann baðar sig means “He bathes” (in a bath), not a shower.
Avoid literal calques like “tekur sturtu”; fara í sturtu is the idiomatic choice.

How are the special letters pronounced here?

þ = unvoiced “th,” like in English “think”: þvær ~ “thvair.”
æ = like the vowel in English “eye.”
é (in sér) = roughly “yeh,” so sér ~ “syer.”
í = long “ee,” as in í sturtu.
Stress is always on the first syllable of each word in Icelandic.

Does eftir always take the accusative?

With the temporal meaning “after,” eftir takes the accusative: eftir viku, eftir fundinn.
With other meanings it can take the dative, e.g., “according to”: eftir lögum (“according to the law”) or authorship: bók eftir Jón (“a book by Jón”).
In eftir vinnu, the form vinnu happens to be the same in acc/dat singular, but here the meaning is temporal (“after work”).