Udomila sam mačku koja mjauče dok je kupam.

This is the Serbian past tense, built with:

• a past participle: udomila
• the auxiliary sam = am

So literally it is something like adopted am.

In Serbian, short unstressed words like sam often go in the second position of the clause, so Udomila sam is very natural.
You can also say Ja sam udomila, where sam is again in second position, after ja.

Also, udomila shows that the speaker is female. A male speaker would say udomio sam.

Udomiti means to adopt or more literally to give/find a home to an animal. In this sentence, udomila sam mačku means I adopted a cat.

It is a perfective verb, so it presents the adoption as a completed event.

Because mačku is the accusative singular form, used for the direct object.

• mačka = a cat, the cat, as the subject form
• mačku = a cat, the cat, as the object form

Compare:

• Mačka mjauče = The cat is meowing
• Kupam mačku = I am bathing the cat

Because inside the relative clause, the cat is the subject of mjauče.

• koja = nominative feminine singular
• koju = accusative feminine singular

Here the structure is basically:

• mačku = the object in the main clause
• koja mjauče = that/which meows, where the cat becomes the subject of the relative clause

So the pronoun must be koja, not koju.

It means that, which, or sometimes who, depending on the noun it refers to.

Here koja refers to mačku, so mačku koja mjauče means the cat that meows.

In Serbian, this kind of relative pronoun agrees with the noun it refers to in gender and number, so because mačka is feminine singular, we get koja.

Because the past tense only applies to the main action: I adopted a cat.

The relative clause koja mjauče dok je kupam describes the cat’s behavior. Serbian, like English, can do this:

• I adopted a cat that meows
• Udomila sam mačku koja mjauče

So the cat’s meowing is presented as a current or habitual characteristic, not as part of the completed adoption event.

Here dok means while.

So:

• dok je kupam = while I bathe it/her or while I am bathing it/her

It introduces an action happening at the same time as the meowing.

Here je is a short unstressed object pronoun meaning her or it, referring to mačku.

So literally:

• dok je kupam = while her/it I bathe

But in natural English, that becomes while I bathe her/it.

This je is not the verb is here. It is the object pronoun.

Because Serbian short pronouns, like short auxiliaries, are clitics. Clitics usually appear very early in the clause, often in the second-position area.

That is why Serbian says:

• dok je kupam

rather than something more English-like with the object after the verb.

This early placement is normal and very important in sounding natural.

It can mean both, depending on context.

In Serbian present tense, kupam can be:

• I bathe
• I am bathing

Here, with dok, it is most naturally understood as while I am bathing her/it.

Also, kupam comes from kupati = to bathe.
Be careful not to confuse it with kupiti = to buy.

Because Serbian usually drops subject pronouns when they are not needed.

The verb form already tells you the person:

• kupam = I bathe / I am bathing
• sam also helps show the first person in the main clause

So ja is unnecessary unless you want emphasis or contrast:

• Ja sam udomila mačku... = I adopted the cat... with extra emphasis on I

Not necessarily. Mačka is the normal everyday word for cat and it is grammatically feminine.

That means other words referring to it also become feminine:

• koja
• je

If you specifically want to say male cat, Serbian often uses mačak.

No. Grammatically, the sentence separates the two ideas:

• Udomila sam mačku = I adopted a cat
• koja mjauče dok je kupam = that meows while I bathe it/her

So the relative clause describes the cat, not the moment of adoption. The cat’s meowing is connected to the bathing, not to the adoption itself.