U zoološkom vrtu ne smijemo hraniti životinje.

Because u can take different cases depending on meaning:

• u + accusative = movement into something
  
  • Idemo u zoološki vrt. = We are going to the zoo.
• u + locative = being in a place
  
  • U zoološkom vrtu... = In the zoo...

In this sentence, there is no movement. It describes a rule that applies in the zoo, so Croatian uses the locative.

That is why:

• zoološki vrt → dictionary form
• u zoološkom vrtu → locative singular

Because adjectives must agree with the noun they describe in:

• gender
• number
• case

Here, vrt is:

• masculine
• singular
• locative

So the adjective zoološki also has to be masculine singular locative:

• zoološki vrt = nominative
• u zoološkom vrtu = locative

So both words change together.

Because vrt is in the locative singular after u when it means in.

So:

• vrt = nominative singular
• u vrtu = locative singular

This is a normal pattern for many masculine nouns in Croatian.

Ne smijemo means we must not, we are not allowed to, or we may not in the sense of lack of permission.

It comes from the verb smjeti = to be allowed to / may.

Here is the form:

• smijemo = we are allowed to
• ne smijemo = we are not allowed to

So the sentence expresses a prohibition, not inability.

Compare:

• Ne smijemo hraniti životinje. = We must not feed the animals.
• Ne možemo hraniti životinje. = We cannot feed the animals.

The second one is about ability or possibility, not permission.

Not necessarily. In Croatian, 1st person plural is often used in a general way, especially for rules, instructions, and public notices.

So ne smijemo can mean:

• we must not
• we are not allowed to
• more generally, people/visitors are not allowed to

In English, a zoo sign would probably say:

• Do not feed the animals
• Visitors must not feed the animals

Croatian often expresses that same idea with ne smijemo.

Because after the modal verb smjeti, Croatian normally uses the infinitive of the main verb.

So:

• smijemo = modal verb
• hraniti = infinitive, to feed

This works like English:

• We may feed
• We must not feed

Other examples:

• Smijem ući. = I may enter.
• Ne smiješ parkirati ovdje. = You must not park here.

Životinje is the direct object, so it is in the accusative plural.

The verb hraniti takes a direct object:

• hraniti životinje = to feed animals

The noun is:

• životinja = animal
• životinje = animals

For this noun, the nominative plural and accusative plural happen to look the same:

• Životinje su gladne. = The animals are hungry.
  Here životinje is nominative plural.
• Hranimo životinje. = We feed the animals.
  Here životinje is accusative plural.

If it were singular, you could see the case change more clearly:

• životinja = nominative singular
• životinju = accusative singular

Yes, that word order is possible, but it can sound a bit different in emphasis.

• U zoološkom vrtu ne smijemo hraniti životinje.
  = In the zoo, we must not feed the animals.
  This starts with the location and feels very natural for a rule or sign.

• Ne smijemo hraniti životinje u zoološkom vrtu.
  = also grammatical, but the location comes later.

Croatian word order is more flexible than English, and moving elements often changes focus rather than basic meaning.

Hraniti is the imperfective verb, and it is the natural choice for a general rule or repeated action.

• hraniti = to feed
• nahraniti = to feed completely / to feed once, to completion

In a prohibition like this, Croatian usually uses hraniti because the rule is broad:

• Do not feed the animals in general.

If you said ne smijemo nahraniti životinje, it would sound more like we are not allowed to feed the animals even once / to give them a feeding, which is less natural for a standard sign.

Yes. Word for word:

• zoološki = zoological
• vrt = garden

So zoološki vrt literally means zoological garden, but in normal English the natural translation is simply zoo.

This is similar to some other languages that use a phrase meaning zoological garden rather than a short single word.

A simple pronunciation guide is:

U zoološkom vrtu ne smijemo hraniti životinje.
oo zo-o-LOSH-kom VUR-too ne SMI-ye-mo HRA-ni-ti zhee-VO-ti-nye

A few useful notes:

• š sounds like sh in shoe
• ž sounds like the s in measure
• smijemo has ije, pronounced roughly smi-ye-mo
• nj in životinje sounds like ny in canyon

The stress patterns can vary a bit by region, but this guide will get you close.