Learn Hardcore Icelandic: Hún gerir köku í ofninum á sunnudögum.

Gera is the infinitive form, meaning to do / to make.
Gerir is the present tense, 3rd person singular form (used with hún = she). So Hún gerir means she makes or she is making.
In Icelandic, the present tense covers both English present simple and present continuous, so Hún gerir köku… can mean either She makes a cake… (habitually) or She is making a cake… depending on context.

Why is it köku and not kaka?

The basic dictionary form is kaka (nominative singular). That form is used for the subject of a sentence.
Here, kaka is the direct object of gerir, so it has to be in the accusative case: köku.
Many feminine nouns like kaka change the vowel and ending in the accusative:

nominative: kaka
accusative: köku

Where is the word “a” in “a cake”? Why is there no separate word for it?

Icelandic has no indefinite article (no equivalent of English a / an).
So köku by itself can mean a cake, one cake, or just cake in general, depending on context.
Definiteness is shown with endings, not with a separate word. For example, the cake (as an object) would be kökuna:

Hún gerir köku = She makes a cake.
Hún gerir kökuna = She makes the cake.

What exactly does í ofninum mean, and why that form?

Í means in / into, and with static location (in the oven) it takes the dative case.
The noun ofn (oven) in dative singular is ofni. When you add the definite ending to say the oven, it becomes ofninum (dative singular definite).
So:

í ofni = in an oven (no article)
í ofninum = in the oven (definite)
If you used í ofninn (accusative), it would mean into the oven (movement), not in the oven.

Why is it á sunnudögum in the plural? What would singular forms mean?

Á sunnudögum is dative plural of sunnudagur (Sunday) and means on Sundays in general, a habitual action.
With days of the week, Icelandic often uses dative plural to express something that regularly happens on that day.
Other common options are:

á sunnudaginn (accusative singular definite) = on Sunday (a specific upcoming/mentioned Sunday)
á sunnudegi (dative singular) = on a Sunday / on Sunday in a more narrative or one-time sense
So á sunnudögum specifically suggests a repeated action, like a routine.

Why does sunnudagur change to sunnudögum?

The base form is sunnudagur (nominative singular).
To make the dative plural, the ending changes to -um, and the stem vowel also changes slightly: sunnudögum.
So the key pattern is:

nominative plural: sunnudagar
dative plural: sunnudögum
In á sunnudögum, á requires the dative for this habitual time expression, so the plural dative sunnudögum is used.

Can I change the order of í ofninum and á sunnudögum?

Yes, you can change the order of the place and time phrases, and the sentence will still be correct. For example:

Hún gerir köku á sunnudögum í ofninum.
This usually puts a bit more emphasis on the time (on Sundays) instead of the place.
You can also front the time expression:
Á sunnudögum gerir hún köku í ofninum.
The main rule is that the finite verb (gerir) stays in second position in the clause, but time and place phrases can move around.

How do I make this sentence negative? Where does ekki go?

The basic pattern is: subject – finite verb – ekki – rest of the sentence.
So: Hún gerir ekki köku í ofninum á sunnudögum. = She does not make a cake in the oven on Sundays.
Other word orders for the adverbials are also possible, but ekki normally comes right after gerir in a simple main clause.

Is gera köku the most natural way to say make a cake in Icelandic?

Gera köku is correct and understandable; gera is a very general verb meaning do / make.
However, for baking, Icelandic speakers often prefer more specific verbs:

baka köku = literally bake a cake (very natural if you mean actually baking)
búa til köku = make a cake (emphasizes putting it together / preparing it)
So you might hear Hún bakar köku í ofninum á sunnudögum at least as often as Hún gerir köku….

Can hún refer to a thing, or only to a woman?

Hún is the 3rd person singular feminine pronoun. It can mean she (a female person), but it can also mean it when referring to a feminine noun.
For example, if you were talking about kakan (the cake, which is feminine), you could later refer to it as hún.
In this sentence, without extra context, Hún is most naturally understood as she (a woman or girl).

What cases are used for each noun or pronoun in this sentence, and why?

Hún – nominative singular (subject of the verb)
köku – accusative singular (direct object of gerir)
ofninum – dative singular definite (object of í expressing location: in the oven)
sunnudögum – dative plural (object of á in this habitual time expression: on Sundays)

So the sentence shows a typical Icelandic pattern: subject in nominative, direct object in accusative, and time/place phrases in dative after the prepositions í and á.