Мой кузен живёт рядом.

Because мой has to agree with кузен.

• кузен is a masculine singular noun
• so the possessive adjective must also be masculine singular nominative
• that form is мой

Compare:

• мой брат = my brother
• моя сестра = my sister
• моё письмо = my letter

Here, Мой кузен means my cousin.

It exists, but in everyday Russian, двоюродный брат is often more common for a male cousin.

So:

• кузен = male cousin
• кузина = female cousin
• двоюродный брат = male cousin
• двоюродная сестра = female cousin

A learner should recognize кузен, but in ordinary speech many Russians would more naturally say двоюродный брат.

Живёт is the 3rd person singular present-tense form of жить (to live).

So:

• я живу = I live
• ты живёшь = you live
• он/она живёт = he/she lives

In this sentence, мой кузен is a singular subject, so живёт means lives.

Yes. In Russian, ё is often written as е in normal texts, even though the pronunciation is still yo.

So:

• живёт = fully marked spelling
• живет = very common unmarked spelling

They mean the same thing, and both are read as zhivYOT, not zhivet.

A useful rule: when you see ё, it is always stressed.

Рядом means nearby, close by, or not far away.

In this sentence:

• Мой кузен живёт рядом. = My cousin lives nearby.

It does not necessarily mean next door specifically. It just means he lives close.

Also, рядом can be used with с + instrumental to mean next to / beside something:

• рядом с домом = next to the house
• рядом со мной = next to me

But here it stands alone and simply means nearby.

Because Russian has no articles.

English distinguishes:

• a cousin
• the cousin
• my cousin

Russian does not use separate words for a or the. Context does that job.

So мой кузен is just my cousin, with no article needed.

Yes. Кузен is in the nominative case, because it is the subject of the sentence.

The basic structure is:

• Мой кузен = the subject
• живёт = the verb
• рядом = adverb

Since кузен is doing the action of living, it stays in the nominative.

Yes. Russian word order is more flexible than English, although the neutral version here is:

• Мой кузен живёт рядом.

You could also say:

• Рядом живёт мой кузен.

That still means basically the same thing, but the focus changes. This version emphasizes nearby first, almost like:

• Nearby lives my cousin.

So the original sentence is the most straightforward, neutral order.

A rough pronunciation guide is:

• Мой → moy
• кузен → koo-ZEN
• живёт → zhi-VYOT
• рядом → RYA-dəm

Stress:

• кузЕн
• живЁт
• рЯдом

So the whole sentence is roughly:

moy koo-ZEN zhi-VYOT RYA-dəm

Yes, if the context already makes it clear whose cousin you mean.

• Кузен живёт рядом. = The cousin / a cousin lives nearby.
• Мой кузен живёт рядом. = My cousin lives nearby.

Including мой makes the relationship explicit. Without it, the sentence is less specific and depends more on context.