Mi bismo pozvali i komšiju, jer on živi blizu nas.

Yes. In Serbian, the subject pronoun is often omitted because the verb form already shows the person and number.

So:

• Bismo pozvali... = We would invite...
• Mi bismo pozvali... = also We would invite..., but with extra emphasis or contrast

Using mi can suggest something like:

• As for us, we would invite...
• We would invite him, maybe unlike someone else

So mi is not strictly necessary here.

This is the conditional form, meaning would invite.

It is made from:

• bismo = the conditional auxiliary for we
• pozvali = the l-participle of pozvati

So:

• mi bismo pozvali = we would invite

A useful pattern is:

• ja bih
• ti bi
• on/ona/ono bi
• mi bismo
• vi biste
• oni/one/ona bi

Then you add the participle:

• bih pozvao / pozvala
• bismo pozvali / pozvale

Note: pozvali is masculine/mixed plural. If the speakers were an all-female group, it would be mi bismo pozvale.

Because Serbian does not form the conditional with an infinitive the way English does.

English:

• we would invite

Serbian:

• mi bismo pozvali

So after bih/bi/bismo/biste, Serbian uses the l-participle, not the infinitive.

That is why you get:

• bismo pozvali not
• bismo pozvati

Because komšiju is the accusative singular form, and here komšija is the direct object of pozvati.

• nominative: komšija = the neighbor
• accusative: komšiju = the neighbor as the object

This happens because komšija refers to a person, and masculine animate nouns usually change form in the accusative.

Compare:

• Komšija živi blizu nas. = The neighbor lives near us.
• Pozvali bismo komšiju. = We would invite the neighbor.

Here i means also / too.

So:

• pozvali i komšiju = invite the neighbor too / also invite the neighbor

It adds the idea that the neighbor is included in addition to other people.

Depending on context, i can also mean and, but here it clearly means also/too.

This is about aspect, which is very important in Serbian.

• pozvati = perfective
• zvati = imperfective

In this sentence, the idea is a single complete action: inviting someone. For that, Serbian normally uses the perfective verb pozvati.

Very roughly:

• pozvati = to invite / to call once, as a complete act
• zvati = to be calling / to call habitually or repeatedly

So bismo pozvali sounds natural for we would invite in one specific situation.

Yes. Jer means because and is very common.

So:

• jer on živi blizu nas = because he lives near us

Another common way to say because is zato što.

For example:

• Pozvali bismo i komšiju, jer živi blizu nas.
• Pozvali bismo i komšiju, zato što živi blizu nas.

Both are natural. Jer is shorter and very common in everyday speech.

Yes, Serbian can often leave it out.

So both of these are natural:

• ...jer on živi blizu nas
• ...jer živi blizu nas

Since živi already tells us it is he/she/it lives, the pronoun is not always needed.

Including on can give a bit more clarity or emphasis, especially if the speaker wants to stress he specifically.

Because blizu normally goes with the genitive.

The pronoun mi has these forms:

• nominative: mi = we
• genitive/accusative: nas
• dative/instrumental/locative: nama

After blizu, Serbian uses nas:

• blizu nas = near us

Other similar expressions also often use the genitive, for example:

• pored nas = beside us
• kod nas = at our place / with us

So blizu nama would not be the standard form here.

Because the second clause states a present fact or general situation:

• he lives near us

The main clause is conditional:

• We would invite...

The reason clause gives the current reason:

• because he lives near us

This is normal in both Serbian and English. The tenses do not have to match mechanically; each clause uses the tense that fits its own meaning.

Yes. Serbian word order is fairly flexible, and changing it usually changes the emphasis rather than the basic meaning.

For example, you may hear:

• Mi bismo pozvali i komšiju, jer on živi blizu nas.
• I komšiju bismo pozvali, jer on živi blizu nas.
• Mi bismo i komšiju pozvali, jer živi blizu nas.

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

• starting with i komšiju emphasizes the neighbor too
• keeping mi bismo near the start sounds neutral
• omitting on sounds slightly less emphatic

So the given sentence is natural, but not the only possible word order.