Bićemo prijatelji zauvek.

Bićemo means we will be.

It is the 1st person plural future tense of biti (to be):

• ja → biću = I will be
• ti → bićeš = you will be
• on/ona/ono → biće = he/she/it will be
• mi → bićemo = we will be
• vi → bićete = you will be
• oni/one/ona → biće = they will be

So in this sentence, bićemo already tells you the subject is we.

Because Serbian often drops subject pronouns when they are already clear from the verb ending.

Here, the ending -mo in bićemo already shows we, so mi is not necessary.

• Bićemo prijatelji zauvek. = We will be friends forever.
• Mi ćemo biti prijatelji zauvek. = also correct, but mi adds emphasis, like we in particular

Serbian does this much more often than English.

Both ideas exist, but bićemo is the natural choice here because ćemo is a clitic. Clitics usually do not stand at the very beginning of a sentence.

So:

• Bićemo prijatelji zauvek. = natural
• Mi ćemo biti prijatelji zauvek. = also natural
• Zauvek ćemo biti prijatelji. = also natural

But:

• Ćemo biti prijatelji zauvek. = not normal as a sentence by itself

So when the sentence begins directly with we will be, Serbian commonly uses the fused form bićemo.

The dictionary form is biti (to be).

This verb is irregular, so its future forms are not completely predictable from the infinitive in a simple way. You should learn biću, bićeš, biće, bićemo, bićete, biće as a set.

So in this sentence:

• biti = to be
• bićemo = we will be

Because the subject is we, and we means more than one person.

The singular is:

• prijatelj = friend

The plural is:

• prijatelji = friends

So:

• Biću prijatelj. = I will be a friend.
• Bićemo prijatelji. = We will be friends.

It is nominative plural.

After biti when you say that someone is / will be something, Serbian often uses the nominative for the predicate noun:

• On je student. = He is a student.
• Oni su prijatelji. = They are friends.
• Bićemo prijatelji. = We will be friends.

So prijatelji is not the object here. It describes what we will be.

Zauvek means forever.

It is an adverb, so it tells you for how long or in what way something happens.

In this sentence, it modifies the whole idea:

• Bićemo prijatelji zauvek. = We will be friends forever.

It does not change form here.

Yes. Serbian word order is fairly flexible, and different orders can shift the emphasis.

For example:

• Bićemo prijatelji zauvek. = neutral, very natural
• Zauvek ćemo biti prijatelji. = emphasis on forever
• Prijatelji ćemo biti zauvek. = possible, but more marked or poetic

So the original sentence is a very normal way to say it, but it is not the only possible order.

In standard Ekavian Serbian, yes: zauvek.

You may also see zauvijek in other varieties, especially Ijekavian forms used in Croatian, Bosnian, Montenegrin, and also some Serbian usage.

So:

• zauvek = common Serbian Ekavian form
• zauvijek = Ijekavian counterpart

Both mean forever.

A rough English-friendly guide is:

• bićemo ≈ BEE-cheh-mo

The letter ć is a soft Serbian sound, somewhere close to a very soft ch. It is softer than English ch in chair.

A rough pronunciation of the whole sentence would be:

• Bićemo prijatelji zauvek
• ≈ BEE-cheh-mo pree-YAH-teh-lyee zah-OO-vek

That is only an approximation, but it can help at the beginning.