Dok ja perem sudove, moja sestra čisti sto.

Here dok means while. It introduces an action happening at the same time as another action.

So:

• Dok ja perem sudove = While I wash/am washing the dishes
• moja sestra čisti sto = my sister cleans/is cleaning the table

In other contexts, dok can also mean until, so you always understand it from the sentence.

Yes, it could be omitted.

Serbian often leaves out subject pronouns because the verb form already shows who is doing the action.

So both are possible:

• Dok ja perem sudove, moja sestra čisti sto.
• Dok perem sudove, moja sestra čisti sto.

Including ja gives a little more emphasis or contrast, something like:

• while I wash the dishes, my sister cleans the table

It is not wrong at all; it just sounds slightly more explicit.

Perem is the 1st person singular present tense of prati/prati se in the sense to wash.

Here is the present tense of prati:

• ja perem = I wash / am washing
• ti pereš = you wash
• on/ona/ono pere = he/she/it washes
• mi peremo = we wash
• vi perete = you wash
• oni/one/ona peru = they wash

So ja perem is the correct form for I wash / I am washing.

Sudove is the accusative plural of sudovi.

The noun sud means dish / piece of tableware, and sudovi means dishes. Because it is the direct object of perem (wash what?), Serbian uses the accusative case.

• nominative plural: sudovi
• accusative plural: sudove

So:

• perem sudove = I wash the dishes

In standard Serbian, sto means table.

The word stol exists in some other South Slavic varieties, but in Serbian sto is the normal word.

Also, in this sentence sto is the accusative singular, but for many masculine inanimate nouns, the accusative singular looks the same as the nominative singular.

So:

• nominative: sto
• accusative: sto

That is why you see:

• čisti sto = cleans the table

Because moja sestra is the subject of the second clause, so it is in the nominative case.

• moja = my
• sestra = sister

Since sestra is feminine singular, the possessive adjective must agree with it:

• moj brat = my brother
• moja sestra = my sister
• moje dete = my child

So moja sestra is simply the normal nominative form meaning my sister.

Čisti is the 3rd person singular present tense of čistiti, meaning to clean.

Present tense:

• ja čistim = I clean / am cleaning
• ti čistiš = you clean
• on/ona čisti = he/she cleans / is cleaning

Since the subject is moja sestra (my sister), Serbian uses ona-type agreement, so:

• moja sestra čisti = my sister cleans / is cleaning

They can mean either, depending on context.

In Serbian, the present tense often covers both:

• I wash
• I am washing

and

• she cleans
• she is cleaning

Because of dok (while), the sentence strongly suggests ongoing simultaneous actions, so in English the most natural translation is usually:

• While I am washing the dishes, my sister is cleaning the table.

But grammatically, Serbian still just uses the present tense forms perem and čisti.

Because the sentence describes actions in progress or happening simultaneously, and Serbian normally uses imperfective verbs for that.

• prati / perem = imperfective to wash
• čistiti / čisti = imperfective to clean

Imperfective verbs are used for:

• ongoing actions
• repeated/habitual actions
• background actions
• simultaneous actions

That fits perfectly with dok in this sentence.

If you used perfective verbs here, the meaning would change and usually sound less natural for simple simultaneous ongoing actions.

Because Dok ja perem sudove is a subordinate clause placed before the main clause.

The structure is:

• subordinate clause: Dok ja perem sudove
• main clause: moja sestra čisti sto

When that time clause comes first, Serbian normally separates it with a comma:

• Dok ja perem sudove, moja sestra čisti sto.

This is very similar to English:

• While I am washing the dishes, my sister is cleaning the table.

Yes, Serbian word order is fairly flexible, though some versions are more neutral than others.

The given sentence is very natural:

• Dok ja perem sudove, moja sestra čisti sto.

You could also say:

• Moja sestra čisti sto dok ja perem sudove.
• Dok perem sudove, moja sestra čisti sto.

The meaning stays basically the same, but word order can change the focus slightly.

For example:

• putting ja in the sentence can emphasize I
• putting moja sestra first can emphasize my sister

Not exactly.

• dok usually emphasizes simultaneous duration: while
• kada usually means when

In this sentence, dok is the better choice because two actions are happening at the same time over a stretch of time:

• Dok ja perem sudove, moja sestra čisti sto.

If you used kada, it would sound more like when as a time point or occasion, not as naturally while.

Serbian has no articles like English a/an/the.

So sto can mean:

• a table
• the table

You understand which one is meant from context.

In this sentence, English would normally say the table, but Serbian simply says sto.

The same is true for sudove:

• sudove = dishes / the dishes

There is no separate article word.

Yes.

Because Serbian present tense with imperfective verbs can also describe repeated or typical actions, the sentence could mean either:

• a scene happening now
  or
• something that generally happens

So it could mean:

• While I’m washing the dishes, my sister is cleaning the table. or
• While I wash the dishes, my sister cleans the table.

Context tells you which reading is intended.

Because Serbian does not need a special progressive construction like English am washing.

English often marks ongoing action with:

• am/is/are + -ing

Serbian usually just uses the normal present tense of an imperfective verb:

• perem = I wash / I am washing
• čisti = she cleans / she is cleaning

So dok helps create the idea of while, but the verbs themselves stay in the regular present tense.