Poslije kiše puž je ostao na zidu pokraj vrata.

Because poslije means after and it normally requires the genitive case.

So:

• kiša = rain
• kiše = of rain / of the rain (genitive singular)

That gives:

• poslije kiše = after the rain or after rain

Croatian has no articles, so context decides whether English should use the or not.

Yes. Nakon kiše also means after the rain.

Both are common:

• poslije kiše
• nakon kiše

In many contexts they are interchangeable. Poslije often feels a bit more everyday and conversational, while nakon can sound a little more neutral or formal, but the difference is not huge here.

Because Croatian past tense is often formed with:

• the auxiliary biti in the present tense
• plus a past participle

Here:

• je = is/has as an auxiliary, from biti
• ostao = past participle of ostati

So:

• puž je ostao = the snail stayed / remained

English usually translates this with a simple past, even though Croatian uses a two-part past form.

Ostao is the masculine singular past participle of ostati (to stay, remain).

It agrees with the subject:

• puž is masculine singular
• so the verb form is ostao

Compare:

• puž je ostao = the snail stayed
• mačka je ostala = the cat stayed
• dijete je ostalo = the child stayed

So the ending changes to match the gender and number of the subject.

Because na can take different cases depending on meaning:

• na + locative = location, on
• na + accusative = movement toward a surface, onto

Here the snail is already located there, so Croatian uses the locative:

• na zidu = on the wall

If you were talking about movement onto the wall, you would use:

• na zid = onto the wall

So:

• Puž je na zidu. = The snail is on the wall.
• Puž je puzao na zid. = The snail crawled onto the wall.

Pokraj means beside, next to, or by.

It normally takes the genitive case.

So:

• pokraj vrata = beside the door / next to the door / by the door

It is similar to:

• kraj
• pored

All of these can often be translated as next to or beside, though usage can vary a little by style and region.

That is a very common learner question. Vrata is a plural-only noun in Croatian when it means door.

So even when English says door in the singular, Croatian usually says vrata.

Examples:

• Vrata su otvorena. = The door is open.
• Stojim pokraj vrata. = I am standing by the door.

This is just something you have to get used to. Grammatically, it behaves like a plural noun.

Because some forms of vrata are identical.

After pokraj, Croatian needs the genitive. The genitive form here is also vrata, so the word does not visibly change.

A useful comparison:

• vrata = nominative/accusative/genitive
• vratima = dative/locative/instrumental

So in this sentence:

• pokraj vrata = beside the door

Even though the case changed, the form happens to look the same.

Not completely. Croatian word order is more flexible than English word order because case endings show grammatical relationships.

This sentence is a natural way to say it:

• Poslije kiše puž je ostao na zidu pokraj vrata.

But other orders are also possible, depending on emphasis:

• Puž je poslije kiše ostao na zidu pokraj vrata.
• Na zidu pokraj vrata puž je ostao poslije kiše.

The version you were given starts with Poslije kiše to set the time first: after the rain.

Because Croatian does not have articles like English the and a/an.

So:

• puž can mean a snail or the snail
• zidu can mean a wall or the wall
• vrata can mean a door or the door

Context tells you which English article makes sense.

That is why one Croatian sentence can correspond to slightly different English versions.

It can mean both, depending on context.

The verb ostati often means:

• to stay
• to remain
• sometimes even to be left

In this sentence, remained is often a very natural translation:

• After the rain, the snail remained on the wall by the door.

But stayed can also work well:

• After the rain, the snail stayed on the wall by the door.

So the exact English choice depends on the tone and the context.

Because they give two different pieces of location information:

• na zidu = on the wall
• pokraj vrata = by/next to the door

Together they narrow the place down more precisely:

• the snail was on the wall
• specifically, by the door

This kind of stacking of place expressions is very normal in Croatian.