Learn Hardcore Indonesian: Saya bertanya kapan kereta berikutnya tiba.

In Indonesian, modifiers usually come after the noun. So you say kereta berikutnya (next train). Putting berikutnya before the noun is not standard in this meaning.

What does the -nya in berikutnya do?

Here -nya helps turn berikut into “the next (one)” and adds a definite feel (like English “the”). It doesn’t mean “his/her/its” here. Berikut without -nya is used in set phrases like berikut ini (as follows) or as “including,” while berikutnya means “next.”

Do I need akan to show the future: kapan kereta berikutnya (akan) tiba?

No. Indonesian doesn’t require a future marker. Both are fine:

kapan kereta berikutnya tiba (natural, especially for schedules)
kapan kereta berikutnya akan tiba (adds an explicit future sense)

Should I use kapan or jam berapa?

kapan = when (general).
jam berapa = at what time (clock time). If you want the exact time: Saya bertanya jam berapa kereta berikutnya tiba.

What’s the difference between tiba, datang, and sampai?

tiba: arrive (formal/neutral; common for transport schedules).
datang: come/arrive (very common in speech).
sampai (di …): reach/arrive at a place; also means “until.”
Your sentence with tiba is perfect for a timetable context; datang also works: kapan kereta berikutnya datang.

Why is there no word like “that” before the embedded question?

Indonesian doesn’t use a complementizer here. You say Saya bertanya kapan …, not Saya bertanya bahwa kapan … (incorrect). Use apakah only for yes–no questions: Saya bertanya apakah kereta berikutnya sudah tiba.

Is the word order inside kapan … fixed? Can I say kapan tiba kereta berikutnya?

Both are acceptable:

kapan kereta berikutnya tiba (very common)
kapan tiba kereta berikutnya (also fine; a bit more formal/literary in feel)

Can I use kapankah in this sentence?

Kapankah is a more formal/emphatic form used in direct questions: Kapankah kereta berikutnya tiba?
Inside your sentence (an embedded question), kapan is more natural: Saya bertanya kapan …. Saya bertanya kapankah … can sound overly formal or bookish.

Is bertanya transitive? How is it different from menanyakan or tanya?

bertanya is intransitive:
- Saya bertanya kapan …
- Saya bertanya kepada petugas.
menanyakan is transitive (ask about something):
- Saya menanyakan jadwal kereta kepada petugas.
tanya (base form) is common in speech:
- Saya tanya kapan kereta berikutnya tiba. (informal)
  Avoid menanya in everyday use; menanyakan is the natural transitive form.

How do I say who I asked? Do I use kepada or pada?

Use kepada (more formal/standard) or pada (also common) with people:

Saya bertanya kepada/pada petugas kapan kereta berikutnya tiba. Colloquial: tanya sama petugas. Don’t use dengan for this meaning.

Is kereta enough, or should I say kereta api?

Both are correct. Kereta is what people commonly say in conversation. Kereta api (literally “train”) is also fine and appears in official names/signage (e.g., PT Kereta Api Indonesia).

Why not bertanya tentang kapan …?

Don’t put tentang directly before a wh- clause. Use either:

Saya bertanya kapan kereta berikutnya tiba. (wh-clause) or
Saya bertanya tentang jadwal kereta. (noun phrase)

Can I report it as direct speech instead of an embedded question?

Yes:

Indirect (what you have): Saya bertanya kapan kereta berikutnya tiba.
Direct: Saya bertanya: Kapan kereta berikutnya tiba?

Is Aku OK instead of Saya? Any politeness tips?

Saya = neutral/formal; safe with strangers or staff.
Aku = informal/intimate.
Polite ways to ask include: Permisi, Maaf, Boleh saya tahu jam berapa kereta berikutnya tiba?, Tolong beri tahu saya …

How do I mark that I asked earlier (past), since there’s no tense?

Add time words:

Tadi/Barusan saya bertanya kapan kereta berikutnya tiba.
Kemarin saya bertanya …
Indonesian uses context and time adverbs rather than verb tense.