Moj otac ide na posao svako jutro.

Because otac is a masculine singular noun, and the possessive adjective moj has to agree with it in gender, number, and case.

So in the nominative singular:

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

Here, otac is the subject of the sentence, so nominative is used, and the correct form is moj.

Because otac is the subject of the sentence, so it stays in the nominative case.

In Moj otac ide na posao svako jutro, the father is the one doing the action, so Serbian uses nominative:

• otac = father, as subject

You would see oca in other roles, for example:

• Vidim oca. = I see my father.
  Here oca is in the accusative, because it is the direct object.

Ide is the 3rd person singular present tense of the verb ići = to go.

Since the subject is moj otac = my father (he), the verb has to match he/she/it:

• ja idem = I go
• ti ideš = you go
• on/ona ide = he/she goes

So:

• Moj otac ide = My father goes

Because this sentence describes a habitual/repeated action: something he does regularly every morning.

In Serbian, repeated or ongoing actions usually use the imperfective verb ići:

• Moj otac ide na posao svako jutro. = My father goes to work every morning.

A perfective verb like otići is more about a single completed departure:

• Moj otac je otišao na posao. = My father left for work / has gone to work.

So ide is the natural choice for a routine.

Because Serbian idiomatically says ići na posao for to go to work.

Even though English uses to, Serbian often chooses prepositions in ways that have to be learned as fixed expressions. With posao, the normal phrase is:

• ići na posao = to go to work
• biti na poslu = to be at work

By contrast, u is more common with enclosed places or institutions:

• ići u školu = to go to school
• ići u grad = to go into town / to the city

So na posao is just the standard Serbian expression.

Because na posao shows movement toward a destination, while na poslu shows location.

With na:

• accusative = motion toward
• locative = location

So:

• ide na posao = he goes to work
• je na poslu = he is at work

Here the verb ide expresses movement, so posao takes the accusative singular form posao.

Svako jutro means every morning, and it is a common Serbian time expression.

• jutro = morning
• svako = every, each

The form svako agrees with jutro, which is neuter singular.

This expression works a bit like an adverb in the sentence:

• Moj otac ide na posao svako jutro. = My father goes to work every morning.

Yes. Svako jutro and svakog jutra can both mean every morning.

Very roughly:

• svako jutro is a very common neutral way to say every morning
• svakog jutra is also common and can sound slightly more like each morning

In everyday use, both are natural:

• Moj otac ide na posao svako jutro.
• Moj otac ide na posao svakog jutra.

A learner should recognize both.

Yes, you could omit moj in some contexts, but the meaning and tone change a little.

• Moj otac clearly means my father
• Otac means father, and depending on context it can sound more general, more definite from context, or slightly more literary/formal

In normal speech, Serbian often omits possessives when the relationship is obvious, but moj otac is perfectly natural and very clear.

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

So Serbian simply says:

• moj otac = my father
• na posao = to work

Whether something is definite or indefinite is usually understood from:

• context
• word order
• pronouns like moj, taj, jedan, etc.

So there is nothing missing here; this is normal Serbian.

Yes. Serbian word order is fairly flexible, although some orders sound more natural depending on emphasis.

The original sentence:

• Moj otac ide na posao svako jutro.

Other possible versions:

• Moj otac svako jutro ide na posao.
• Svako jutro moj otac ide na posao.

These all mean roughly the same thing, but the focus shifts a little:

• svako jutro at the start emphasizes every morning
• placing it later sounds more neutral

The original sentence is completely natural.

Because adjectives and adjective-like words in Serbian agree with the noun they describe.

Here:

• jutro is neuter singular
• so svaki becomes svako

Compare:

• svaki dan = every day (dan is masculine)
• svaka noć = every night (noć is feminine)
• svako jutro = every morning (jutro is neuter)

This agreement pattern is very important in Serbian.

It normally describes a routine.

Even though the verb is in the present tense, the phrase svako jutro makes it clear that this is something habitual:

• Moj otac ide na posao svako jutro. = My father goes to work every morning.

Without a time phrase, ide could sometimes mean is going depending on context, but here the sentence is clearly about repeated action.