Learn Hardcore German: In den Sommerferien kochte meine Tante jeden Abend Kartoffeln für die hungrigen Passagiere auf ihrem kleinen Boot. - During the summer holidays my aunt cooked potatoes every evening for the hungry passengers on her small boat.

Sommerferien is one of those German nouns that exists only in the plural form (you say die Ferien, never a singular).
Because in is a two-way preposition, it takes the dative case when indicating a time period (no movement).
The dative plural of die Ferien is den Ferien (with an extra -n). Hence In den Sommerferien means “during the summer holidays.”

Why does the finite verb kochte come immediately after In den Sommerferien, even before the subject meine Tante?

German main clauses follow the Verb-second (V2) rule:

Whatever element (subject, adverbial, object) you put first, the finite verb must be in the second position.
Here In den Sommerferien is fronted, so the verb kochte stays in position 2, and then the subject meine Tante follows.

What case is meine Tante in, and why isn’t there a definite article before it?

meine Tante is the subject of the sentence, so it’s in the nominative case.
The word meine is a possessive pronoun (meaning “my”) that replaces the need for the definite article die. In German you don’t say die meine Tante; meine Tante already includes the determiner.

Why is jeden Abend used instead of just abends, and where do time expressions like this usually go in a German sentence?

jeden Abend translates to “every evening.” You could also say abends (“in the evenings”), but jeden Abend explicitly highlights each individual evening.
Time expressions typically appear early in the clause. Because we already fronted In den Sommerferien, jeden Abend comes after the subject here.
When you have two time expressions, German often orders them from general to specific:
• In den Sommerferien (broad period)
• jeden Abend (repeated daily)

Why is there no article before Kartoffeln?

Here Kartoffeln (potatoes) is an indefinite, plural direct object describing what is being cooked in general.
In German you can use a bare plural when talking about things in an unspecific or general sense (“potatoes” rather than “the potatoes” or “some potatoes”).

Why is für die hungrigen Passagiere in the accusative case, and why does hungrigen have the -en ending?

The preposition für always takes the accusative case.
Passagiere is a plural noun, so the accusative plural definite article is die.
Adjectives following a definite article take the weak ending -en in all cases except nominative singular. Hence die hungrigen Passagiere.

Why is it auf ihrem kleinen Boot (dative) instead of accusative, and why are both ihrem and kleinen ending in -em/-en?

auf is a two‐way preposition; with no movement (static location) it uses the dative case.
Boot is neuter (das Boot). The dative singular of das Boot is dem Boot.
A possessive pronoun like ihr- (“her”) takes the dative ending: ihr → ihrem.
After a determiner (like ihrem), the adjective uses the weak -en ending in every case except nominative singular. Hence kleinen.
→ auf ihrem kleinen Boot

Why is the verb in the simple past tense (kochte) instead of the perfect (hat … gekocht)?

In spoken German you usually use the Perfekt (e.g. hat … gekocht) for past events, except with a few very common verbs.
In written narrative (stories, reports, descriptions of repeated actions) you more often use the Präteritum (simple past).
kochen is a regular verb that works perfectly in the simple past here: kochte.