Poslije kiše iznad brda vidimo dugu, dok je livada još mokra.

Kiše is in the genitive singular of kiša (rain).

The preposition poslije means after, and it normally requires the genitive:

• poslije kiše = after the rain
• poslije škole = after school
• poslije ručka = after lunch

So:

• kiša = rain
• kiše = of rain / after rain, depending on context

In this sentence, poslije kiše functions as a time expression: after the rain.

Brda is also genitive singular, from brdo (hill).

The preposition iznad means above / over, and it also takes the genitive:

• iznad brda = above the hill
• iznad grada = above the city
• iznad kuće = above the house

So in this sentence:

• iznad = above
• brda = of the hill / hill in the genitive after iznad

Together, iznad brda means above the hill.

Croatian often leaves out subject pronouns when they are already clear from the verb form.

Vidimo means we see, because the ending -mo shows 1st person plural:

• vidim = I see
• vidiš = you see
• vidi = he/she/it sees
• vidimo = we see

So mi is not necessary unless you want emphasis or contrast:

• Vidimo dugu. = We see a rainbow.
• Mi vidimo dugu, a oni ne. = We see a rainbow, but they do not.

In your sentence, vidimo by itself is perfectly natural.

Because dugu is the accusative singular form of duga (rainbow).

The verb vidjeti / vidimo takes a direct object, and direct objects are often in the accusative:

• Vidimo dugu. = We see a rainbow.

The noun duga is feminine, so:

• nominative: duga
• accusative: dugu

Compare:

• Duga je lijepa. = The rainbow is beautiful.
  Here duga is the subject, so it is nominative.
• Vidimo dugu. = We see the rainbow / a rainbow.
  Here it is the object, so it is accusative.

Good question. Yes, dugu can also be the feminine accusative form of the adjective dug (long), but here it is clearly a noun meaning rainbow.

Why? Because vidimo dugu on its own naturally means we see a rainbow.
If dugu were an adjective, it would normally need a noun:

• vidimo dugu cestu = we see a long road
• vidimo dugu haljinu = we see a long dress

So in this sentence, dugu is the noun duga in the accusative.

Here dok means while.

It introduces a clause that describes something happening at the same time:

• Vidimo dugu, dok je livada još mokra.
• We see a rainbow, while the meadow is still wet.

In other contexts, dok can sometimes mean until, so learners often notice that it has more than one use. But in this sentence, the meaning is clearly while.

This is because je is a clitic in Croatian.

Clitics are short unstressed words that usually appear in the second position of their clause. In the clause:

• dok je livada još mokra

the first element is dok, so je comes right after it.

This is very normal in Croatian:

• Kad je došao... = When he came...
• Ako se sjećam... = If I remember...
• Dok je livada mokra... = While the meadow is wet...

English speakers often expect livada je, but Croatian word order follows different rules because of clitics.

Because mokra is an adjective describing livada, and it must agree with the noun in gender, number, and case.

• livada = meadow, feminine singular
• mokra = wet, feminine singular nominative

In this clause, livada is the subject, so it is nominative. The adjective mokra matches it:

• Livada je mokra. = The meadow is wet.

If the noun changed, the adjective would change too:

• brdo je mokro = the hill is wet
• trava je mokra = the grass is wet
• putevi su mokri = the roads are wet

Here još means still.

So:

• livada je još mokra = the meadow is still wet

It shows that the wetness continues at that moment.

Još can also mean other things depending on context, such as more or another, but here the meaning is definitely still:

• Još spava. = He/She is still sleeping.
• Još jedan primjer. = One more example.

Croatian has no articles like English the and a/an.

So:

• kiša can mean rain or the rain
• brdo can mean a hill or the hill
• duga can mean a rainbow or the rainbow
• livada can mean a meadow or the meadow

The exact meaning usually comes from context.

So vidimo dugu could be understood as:

• we see a rainbow or
• we see the rainbow

English has to choose an article, but Croatian usually does not.

Yes, Croatian word order is more flexible than English.

The sentence begins with Poslije kiše and iznad brda because these set the scene first:

• after the rain
• above the hill

Then comes the main action:

• vidimo dugu = we see a rainbow

Other word orders are possible, for example:

• Vidimo dugu iznad brda poslije kiše.
• Poslije kiše vidimo dugu iznad brda.

These all express basically the same idea, but the emphasis changes slightly. The version in your sentence sounds natural because it presents the situation in a descriptive, scene-setting way.

Because dok introduces a subordinate clause:

• main clause: Poslije kiše iznad brda vidimo dugu
• subordinate clause: dok je livada još mokra

In standard Croatian, clauses like this are normally separated by a comma.

So the comma helps show the structure:

• We see a rainbow above the hill after the rain, while the meadow is still wet.

Yes, poslije and nakon can both mean after, and both take the genitive.

So you could say:

• Poslije kiše...
• Nakon kiše...

Both are correct. In many contexts they are interchangeable. Poslije often feels a bit more everyday and conversational, while nakon can sound slightly more formal or neutral, but both are very common.

So Nakon kiše iznad brda vidimo dugu... would also be grammatical and understandable.