Learn Hardcore Icelandic: Sturtan er heit í morgun.

sturta = “a shower” (indefinite nominative singular)
sturtan = “the shower” (definite nominative singular)
Since we’re talking about the specific shower (e.g. the one in your house), you use the definite form sturtan.

What is the gender of sturtan, and how can I tell?

Sturta is a feminine noun. Clues:

Many Icelandic feminine nouns end in -a in the indefinite form.
You’ll see agreement on adjectives and articles later (e.g. ein sturta, sturtan).

Knowing the gender is crucial because adjectives, pronouns and past participles must agree with the noun’s gender, number and case.

Why is the adjective heit used instead of heitur?

Heitur (“hot”) must agree with sturtan (feminine, singular, nominative). In the strong declension you get:

Masculine nom sg: heitur
Neuter nom/acc sg: heitt
Feminine nom sg: heit
Because sturtan is feminine, “hot” becomes heit when used predicatively (“the shower is hot”).

Why is the adjective placed after the verb er and not directly after the noun?

In Icelandic, when you describe the subject with a “to be” verb (vera → er), the adjective is predicative. Predicative adjectives always come after the verb.
Compare:

Predicative: Sturtan er heit. (“The shower is hot.”)
Attributive: heit sturta (without “er”, this means “a hot shower”).

What case is í morgun, and why is there no article?

Morgunn (“morning”) is normally masculine. The accusative singular is morgun.
The preposition í can introduce a temporal phrase in the accusative without any article.
í morgun is simply the fixed way to say “this morning.”

Could I say Sturtan var heit í morgun instead? When do I use er vs. var?

Er is present tense of vera (“to be”). You use er if it’s still morning or you’re talking about it right now.
Var is past tense (“was”). You’d say Sturtan var heit í morgun if the morning is over and you want to report it as past.

How would you say “I took a hot shower this morning” in Icelandic?

Ég tók heita sturtu í morgun.
Here:

tók = past of taka (“to take”)
sturtu = accusative feminine singular of sturta (indefinite)
heita = accusative feminine singular of heitur (strong declension attributive)

How is Sturtan er heit í morgun. pronounced? Any tricky sounds?

Approximate IPA: [ˈstʏr.tan ɛr ˈheit iː ˈmɔr.kʏn]
Tips:

u in Icelandic is [ʏ], like German “ü.”
st at the start is [st], not [ʃt].
guttural r is an alveolar trill [r].
j (in morgun) is [j].

What’s the basic word order here? Could I start with Í morgun?

The default is S-V-O with adverbial phrases often after the verb:
Subject – Verb – Predicate – Adverbial.
Example: Sturtan er heit í morgun.
But you can front the time phrase for emphasis:
Í morgun er sturtan heit.
Just remember that if you start with an adverbial, the verb usually follows it, then the subject.