Learn Hardcore Dutch: Na de regen verschijnen er mooie regenbogen in de lucht. - After the rain, beautiful rainbows appear in the sky.

Na is a preposition that introduces a time adverbial and must be followed by a noun phrase (de regen). If you want to introduce a subordinate clause (“after it rained…”), you use the conjunction nadat, as in Nadat het geregend heeft, verschijnen er mooie regenbogen in de lucht.

Why does verschijnen come before er in Na de regen verschijnen er mooie regenbogen?

Dutch is a verb‐second (V2) language. When a time expression (Na de regen) occupies the first position, the finite verb (verschijnen) moves into second position, pushing the dummy subject er to come immediately after the verb.

What role does er play in this sentence?

Er is a pleonastic or dummy subject used in existential constructions. It fills the grammatical subject slot so that the real, indefinite subject (mooie regenbogen) can follow the verb.

Can you omit er here? If not, how could you rephrase the sentence without it?

In the given word order you cannot drop er, because Dutch requires a subject in subject position. To omit er, you must front the real subject. For example:
Mooie regenbogen verschijnen in de lucht na de regen.

Why does mooi gain an -e ending in mooie regenbogen?

In Dutch, adjectives preceding a plural noun always take the -e ending. That’s why mooi becomes mooie before regenbogen.

How is the plural regenbogen formed from regenboog?

Dutch compounds take the plural on their head noun. The head of regenboog (“rain” + “arc”) is boog, whose plural is bogen (the long oo shortens to o in the plural). Hence regenbogen.

Why is in used in in de lucht? Could you use op or boven instead?

In de lucht is the standard expression for “in the sky” or “in the air.”
Op de lucht is ungrammatical here, and boven de lucht would literally mean “above the sky,” which isn’t what you want.