Imam kućnog ljubimca.

Because Serbian verb forms usually already show the subject.

• imam = I have
• imaš = you have
• ima = he/she/it has

So ja (I) is not necessary here. You can say Ja imam kućnog ljubimca, but it sounds more emphatic, like I do have a pet or I have a pet with extra stress on I.

A simple breakdown is:

• Imam = I have
• kućnog = domestic / household / pet-
• ljubimca = pet

Together, kućni ljubimac is the usual Serbian expression for pet, and in this sentence it changes form to kućnog ljubimca because of case.

Because after imam (I have), the direct object goes in the accusative case.

The dictionary form is:

• kućni ljubimac = a pet / pet
  nominative singular

But in Imam kućnog ljubimca, the phrase is the object of the verb, so it must be in the accusative:

• kućnog ljubimca = accusative singular

Good question. In Serbian, masculine animate nouns often have an accusative singular form that looks exactly like the genitive singular.

For example:

• nominative: ljubimac
• genitive: ljubimca
• accusative: ljubimca
  when it refers to an animate being

Since a pet is animate, Serbian uses:

• Imam kućnog ljubimca

So it is accusative in function, even though its form matches the genitive.

The adjective has to agree with the noun in gender, number, and case.

The noun ljubimac is:

• masculine
• singular
• animate
• accusative in this sentence

So the adjective changes too:

• nominative: kućni ljubimac
• accusative masculine animate: kućnog ljubimca

That is why both words change together.

Yes. Kućni ljubimac is the standard and very common expression for pet in Serbian.

Literally, it is something like household beloved animal/creature, but in normal use it simply means pet.

You will often see forms such as:

• kućni ljubimac = pet
• imam kućnog ljubimca = I have a pet
• voli svog kućnog ljubimca = he/she loves his/her pet

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

So Imam kućnog ljubimca can mean:

• I have a pet
• sometimes, depending on context, I have the pet

In this sentence, the most natural translation is I have a pet.

Context usually tells you whether something is indefinite or definite.

Yes, absolutely.

• Imam kućnog ljubimca = neutral, natural
• Ja imam kućnog ljubimca = more emphasis on I

You might use ja if you are contrasting:

• On nema kućnog ljubimca, ali ja imam kućnog ljubimca.
  He doesn’t have a pet, but I do.

So the version without ja is usually the default.

Not as fixed as in English.

The most neutral order is:

• Imam kućnog ljubimca.

But Serbian allows other orders for emphasis or style, for example:

• Kućnog ljubimca imam.

That sounds marked or contrastive, something like:

• A pet, I do have.

For learners, it is best to use Imam kućnog ljubimca unless you have a specific reason to change the order.

A rough guide:

• lj in ljubimca sounds like the lli in million for many English speakers, though not exactly the same.
• ć in kućnog is a soft sound, somewhat like a very soft ch.
• c in ljubimca is pronounced like ts.

Very roughly:

• ljubimca ≈ lyoo-BEEM-tsa
• kućnog ≈ KOOCH-nog, but with a softer ch

Serbian spelling is quite regular, so once you learn the sound values, pronunciation becomes much easier.

The plural would be:

• Imam kućne ljubimce. = I have pets.

Compare:

• singular: Imam kućnog ljubimca.
• plural: Imam kućne ljubimce.

Here:

• kućne = plural accusative form of the adjective
• ljubimce = plural accusative form of ljubimac

This is a useful pattern to learn alongside the singular.