Learn Hardcore German: Nach dem Regen erscheint ein bunter Regenbogen über der Stadt. - After the rain a colorful rainbow appears over the city.

German uses nach as a preposition followed by a noun in the dative case, so we say nach dem Regen. In contrast, nachdem is a subordinating conjunction introducing a full clause (e.g. Nachdem es geregnet hat…); with nachdem the verb goes to the end of that clause. Here we want a simple time‐phrase, not a subordinate clause, so we use nach + dative.

Why is the adjective bunter in ein bunter Regenbogen, and how do German adjective endings work here?

In ein bunter Regenbogen we have:

An indefinite article ein (nominative masculine singular)
The noun Regenbogen (masculine)
The adjective bunt

After an indefinite article in the nominative masculine, the adjective takes the ending -er.
Quick declension chart (nominative singular):
ein bunter (masc.)
eine bunte (fem.)
ein buntes (neut.)
bunte (pl., no article or with definite article)

Why is it über der Stadt instead of über die Stadt, and which case does über take here?

Über is a Wechselpräposition (two‐way preposition).

With movement (direction), it takes accusative (e.g. über die Brücke gehen)
In a static location (position), it takes dative

Here the rainbow is resting over the city (no movement toward it), so we use the dative. Stadt is feminine, dative singular → der Stadt.

Why is the verb erscheint in second position, and why does the subject ein bunter Regenbogen come after it?

In German main clauses the finite verb occupies the second “slot” (V2 rule). You can place different elements in the first slot (time, place, object, etc.):

Nach dem Regen (time – first slot)
erscheint (verb – second slot)
ein bunter Regenbogen (subject – third slot)

If you start with the subject, you’d get Ein bunter Regenbogen erscheint nach dem Regen … The order shifts emphasis but the verb-second rule remains.

What grammatical function does Nach dem Regen serve in this sentence?

Nach dem Regen is an adverbial phrase of time (temporal adverbial). It answers “Wann?” (when?) and is often placed at the beginning of a German sentence for emphasis or clarity.

How would I rewrite this sentence using the subordinate conjunction nachdem?

Use nachdem to introduce a subordinate clause with the verb at the end, then follow with the main clause:
Nachdem es geregnet hat, erscheint ein bunter Regenbogen über der Stadt.

Is a comma needed after Nach dem Regen?

No comma is required after a simple fronted adverbial phrase. Commas are mandatory for subordinate clauses or between main clauses, but not for single adverbials like Nach dem Regen erscheint…