Learn Hardcore Italian: Dopo il lampo è apparso un arcobaleno sopra la collina. - After the flash of lightning a rainbow appeared over the hill.

In Italian dopo can act as a preposition when it’s followed by a noun phrase (here il lampo). You only add che if you introduce a full clause with its own verb.
Examples:

Preposition + noun: Dopo il film siamo usciti.
Conjunction + clause: Dopo che il film è finito, siamo usciti.

Why is Dopo il lampo placed at the beginning of the sentence?

Putting an adverbial expression of time at the start is common to set the scene or emphasize the sequence. Italian word order is quite flexible, so you could also say:
Un arcobaleno è apparso sopra la collina dopo il lampo.
Starting with Dopo il lampo simply highlights when the rainbow appeared.

Do we need a comma after Dopo il lampo?

No, a comma is optional. Italian allows:

Dopo il lampo è apparso…
Dopo il lampo, è apparso…
Use a comma if you want a slight pause or clearer separation, but it’s not grammatically required.

Why is the definite article il used before lampo, but an indefinite un before arcobaleno?

Here il lampo refers to the specific flash that just happened (given information), so it takes the definite article. Un arcobaleno introduces a new occurrence (new information), so it uses the indefinite article un.

Why is the indefinite article un used before arcobaleno, not uno?

In Italian uno is reserved for masculine nouns beginning with z, s + consonant, gn, ps and so on. Since arcobaleno begins with a vowel, the correct form is un.

Why is the past participle apparso used with essere rather than avere?

Apparire is an intransitive verb indicating movement or a change of state (becoming visible), so it forms the passato prossimo with essere. With essere, the past participle agrees in gender and number with the subject (here masculine singular, so apparso).

What tense is è apparso, and could we use a different one (like imperfetto or passato remoto)?

È apparso is the passato prossimo, used for a completed action in the recent past. Alternatives:

Appariva (imperfetto) to describe an ongoing or repeated appearance in the past.
Apparve (passato remoto) in more literary or historical narratives.
In everyday spoken Italian about a recent event, passato prossimo is preferred.

Why is sopra la collina used instead of su or sulla collina?

Both sopra and su can mean on/over, but sopra often emphasizes something directly above or higher than a reference point. Saying sopra la collina highlights that the rainbow was in the sky over the hill. Sulla collina (on the hill) would imply it was physically on top of the hill itself, which doesn’t fit this image as naturally.

Could we replace apparso with comparso, and is there a difference?

Yes. Comparire is a near-synonym of apparire, also taking essere in the passato prossimo (è comparso). Nuance:

Apparire often focuses on the gradual process of coming into view.
Comparire can stress a sudden or unexpected appearance.
In most contexts, though, they’re interchangeable.