Mir je važan u svakoj zemlji.

A very natural way to read it is:

• Mir = the subject
• je = is
• važan = important
• u svakoj zemlji = in every country

So the structure is basically:

subject + je + adjective + place phrase

That is a normal Serbian sentence pattern. Serbian word order is more flexible than English, but this version is straightforward and neutral.

Mir is in the nominative singular, which is the form usually used for the subject of the sentence.

In this sentence, mir is the thing being described as important, so it is the subject:

• Mir = peace
• je važan = is important

So the sentence is saying something about peace.

Because važan agrees with mir.

The noun mir is:

• masculine
• singular
• nominative

So the adjective also has to be:

• masculine
• singular
• nominative

That gives važan.

Compare:

• mir je važan = peace is important
• sloboda je važna = freedom is important
• dete je važno = a child / the child is important

The adjective changes form to match the noun.

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

So:

• mir je važan = peace is important

English also needs is here, and Serbian does too in standard usage.

Because this sentence describes location, not movement.

With u, Serbian usually works like this:

• u + locative = in, inside, within a place
• u + accusative = into a place, showing motion

Here, the idea is in every country, so it is location. That is why Serbian uses the locative:

• u svakoj zemlji = in every country

If you were talking about motion, you would use accusative:

• Idem u svaku zemlju = I am going into / to every country

Because both words are in the locative singular feminine.

The noun zemlja is feminine, and after u in a location meaning, it goes into the locative:

• zemlja → zemlji

The word svaka = every must agree with zemlja, so it also changes:

• svaka → svakoj

So:

• u svakoj zemlji = in every country

This is a good example of agreement inside a noun phrase.

The dictionary form is svaki.

It changes depending on gender, number, and case. In this sentence it matches zemlji, so it becomes svakoj.

Some useful forms are:

• svaki grad = every city
• svaka zemlja = every country
• svako dete = every child

And in the locative singular feminine:

• u svakoj zemlji

Yes. Serbian word order is fairly flexible.

For example, these are also possible:

• U svakoj zemlji mir je važan.
• Važan je mir u svakoj zemlji.

The basic meaning stays very similar, but the focus changes a little.

• Mir je važan u svakoj zemlji sounds neutral.
• U svakoj zemlji mir je važan puts a little more focus on in every country.
• Važan je mir... puts more focus on important or on peace as the key idea.

So the original sentence is a normal, neutral choice.

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

So mir can mean simply peace, without any article. Serbian relies on context instead of articles.

That is very normal:

• Mir je važan = Peace is important
• Zemlja je velika = The country is big / A country is big, depending on context

It is still an adjective, but here it is used in a predicate position, after je.

Compare:

• važan mir = important peace
  adjective directly before the noun

• mir je važan = peace is important
  adjective used after je to describe the subject

So it is an adjective in both cases, just in a different sentence role.

Yes. Bitan can also mean important, and in many contexts it works well.

So:

• Mir je važan u svakoj zemlji
• Mir je bitan u svakoj zemlji

Both are natural. Važan is often a little more neutral and standard for learners, while bitan is also very common in everyday speech.

From both form and agreement.

• svakoj is feminine singular
• zemlji here is the locative singular form of zemlja

So the phrase means in every country, not in all countries.

If you wanted a plural idea, you would build the phrase differently, for example with u svim zemljama = in all countries.

Yes. Mir is a masculine noun, so adjectives that describe it must use masculine forms:

• mir je važan
• mir je trajan
• mir je moguć

That is why you get važan, not važna or važno.