Moja majka je umorna svako veče.

Because moja has to agree with majka in gender, number, and case.

• majka is a feminine singular noun
• so the possessive my must also be feminine singular: moja

Compare:

• moj otac = my father
• moja majka = my mother
• moje dete = my child

So moja is the correct form because majka is feminine.

For the same reason: the adjective must match the noun.

• majka is feminine singular
• so umoran (tired) becomes umorna

Compare:

• On je umoran. = He is tired.
• Ona je umorna. = She is tired.
• Dete je umorno. = The child is tired.

So umorna agrees with majka.

Je is the 3rd person singular present form of the verb biti (to be).

So in this sentence:

• majka je umorna = mother is tired

It works just like is in English.

A few forms of biti in the present tense:

• sam = am
• si = are
• je = is
• smo = are
• ste = are
• su = are

So je is necessary here because Serbian usually uses the verb to be in sentences like this.

In standard Serbian, not in this sentence.

You need je in:

• Moja majka je umorna.

If you remove it:

• Moja majka umorna.

that sounds incomplete or non-standard.

Serbian sometimes omits forms of to be in very informal speech, headlines, or special styles, but learners should treat je as required here.

Svako veče means every evening.

• svako = every (agreeing with a neuter singular noun)
• veče = evening

The noun veče is neuter singular, so svako also appears in the neuter singular form.

This phrase is used adverbially, meaning it tells us when something happens:

• Moja majka je umorna svako veče.
  = My mother is tired every evening.

Yes. Svake večeri is also very natural and means essentially the same thing: every evening.

So both are possible:

• Moja majka je umorna svako veče.
• Moja majka je umorna svake večeri.

The difference is mostly one of expression, not basic meaning.

A learner-friendly way to think about it:

• svako veče = a very common straightforward way to say every evening
• svake večeri = also common, slightly different structure

Both are correct and useful.

No, Serbian word order is fairly flexible.

The sentence:

• Moja majka je umorna svako veče.

could also appear as:

• Moja majka je svako veče umorna.
• Svako veče je moja majka umorna.

These all mean roughly the same thing, but the emphasis changes.

A simple rule for learners:

• Serbian word order is often used to shift focus or emphasis
• the most neutral version here is the original: Moja majka je umorna svako veče.

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

So:

• majka can mean mother or the mother, depending on context
• moja majka naturally means my mother

English requires articles, but Serbian does not.

That is why Serbian sentences often look shorter than their English equivalents.

Yes, you could say that, but the meaning changes slightly.

• Moja majka je umorna svako veče. = My mother is tired every evening.
• Majka je umorna svako veče. = Mother/The mother is tired every evening.

Without moja, you lose the explicit idea of my.
Whether that is still clear depends on context.

So if you specifically want to say my mother, keep moja.

In Serbian, the letter j is pronounced like English y in yes.

So:

• moja sounds roughly like MO-ya
• majka sounds roughly like MY-ka
• je sounds roughly like ye

That is important because English speakers often want to pronounce j like English j, but in Serbian it usually sounds like y.