Pas nosi ogrlicu i šetam ga na povocu.

Because ogrlicu is the accusative singular form of ogrlica.

• ogrlica = nominative singular, the dictionary form
• ogrlicu = accusative singular

In this sentence, ogrlicu is the direct object of nosi (wears / carries), so Serbian uses the accusative:

• Pas nosi ogrlicu. = The dog is wearing a collar.

Since ogrlica is a feminine noun ending in -a, its accusative singular usually changes to -u.

Povocu is the form used after na in the expression na povocu = on a leash.

The base noun is povodac = leash.
After na, Serbian often uses the locative when talking about being on / in / at something in a static sense.

So:

• povodac = leash
• na povocu = on a leash

This is a very common phrase, and it is best learned as a set expression.

The person doing the walking is I, even though ja is not written.

That is because šetam already means I walk / I am walking / I take for a walk. Serbian verb endings usually show the subject, so subject pronouns are often omitted.

So:

• šetam = I walk / I am walking
• implied subject = ja

That means the sentence has two different subjects:

• Pas nosi ogrlicu → the dog is the subject
• i šetam ga na povocu → I am the subject

Serbian allows this without repeating ja, because the verb form makes it clear.

Because ga is the unstressed accusative pronoun meaning him.

Here the dog is the direct object of šetam:

• šetam psa = I walk the dog
• šetam ga = I walk him

Compare:

• on = he
• njega = him, but the stressed / full form
• ga = him, the usual short clitic form

In a normal sentence like this, Serbian usually uses the short form ga.

Because ga is a clitic, and Serbian clitics have special word-order rules.

Very roughly, clitics tend to come near the beginning of their clause, after the first stressed element. In this clause:

• i šetam ga na povocu

the verb šetam is the first stressed word, so ga comes right after it.

That is why šetam ga sounds natural, while ga šetam would not be the normal order here.

Yes, nositi can mean both to carry and to wear, depending on context.

With clothing or accessories, it often means wear:

• nosi kaput = wears a coat
• nosi naočare = wears glasses
• nosi ogrlicu = wears a collar

So in this sentence, Pas nosi ogrlicu means the dog has a collar on.

Yes. Ogrlica can mean something worn around the neck in general, so the exact English word depends on context.

• for a dog: collar
• for a person: often necklace

In this sentence, because the subject is pas, ogrlica clearly means collar.

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

So:

• pas can mean a dog or the dog
• ogrlicu can mean a collar or the collar
• povocu can mean a leash or the leash

The exact meaning depends on context.

In Serbian, šetati can be used transitively, especially when talking about taking a dog for a walk.

So:

• Šetam psa. = I walk the dog / I take the dog for a walk.
• Šetam ga. = I walk him.

If you want to say walk with the dog, you could also use a different structure, such as:

• Šetam sa psom. = I am walking with the dog.

But that is slightly different in focus:

• šetam psa / šetam ga = I am the one taking the dog out
• šetam sa psom = I am walking together with the dog

The sentence is understandable and grammatical, but many speakers might make the subject change a little clearer.

For example:

• Pas nosi ogrlicu, a ja ga šetam na povocu.

Using a here helps show the switch from the dog in the first clause to I in the second clause.

The original sentence is still fine, but a learner might find the version with a ja easier to parse.