Мой друг кивает и улыбается.

Because друг is a masculine singular noun, and мой has to agree with it.

• мой = masculine singular
• моя = feminine singular
• моё = neuter singular
• мои = plural

So:

• мой друг = my friend (male friend, one person)

It is in the nominative case, because it is the subject of the sentence — the person doing the actions.

Here, друг is the one who nods and smiles, so nominative is the expected case.

Because the subject is third person singular: мой друг = he / my friend.

So the verbs are in the present tense, 3rd person singular:

• кивать → кивает
• улыбаться → улыбается

Russian does not need a separate word like English is here. The verb form itself already tells you the tense and person.

The dictionary forms (infinitives) are:

• кивать = to nod
• улыбаться = to smile

So:

• кивает = nods / is nodding
• улыбается = smiles / is smiling

Because the verb is улыбаться, which is a reflexive verb.

The ending -ся (or -сь) is a very common part of many Russian verbs. Sometimes it has a clear reflexive meaning, but often it is just part of the standard verb form.

With улыбаться, you simply have to learn it as:

• улыбаться = to smile

You normally do not use улыбать to mean to smile in standard modern Russian.

It can mean either one, depending on context.

Russian present tense often covers both:

• simple present: nods and smiles
• present continuous: is nodding and smiling

So Мой друг кивает и улыбается can describe:

• a general/habitual action, or
• something happening right now

The wider context tells you which meaning is intended.

Because Russian has no articles.

English says:

• a friend
• the friend
• my friend

Russian just says:

• друг
• мой друг

Whether the meaning is more like a or the depends on context.

Yes. Russian word order is more flexible than English.

The neutral/basic order here is:

• Мой друг кивает и улыбается.

But you could also hear:

• Мой друг улыбается и кивает.
• Кивает и улыбается мой друг.

These alternatives may shift the focus or sound more literary/emphatic, but the core meaning stays similar.

These are imperfective verbs:

• кивать
• улыбаться

That makes sense because the sentence describes an action as ongoing, repeated, or simply in progress.

Also, in Russian, a true present tense form is normally used with imperfective verbs. Perfective verbs do not usually have a normal present-tense meaning; their present-looking forms usually refer to the future.

Compare:

• кивает, улыбается = nods / is nodding, smiles / is smiling
• кивнёт, улыбнётся = will nod, will smile

The stress is:

• Мой друг кива́ет и улыба́ется.

A rough English-friendly pronunciation is:

• moy drook kee-VA-yet ee oo-ly-BA-yet-sya

Two helpful notes:

• друг ends with a sound closer to k than English g, because final voiced consonants are devoiced in Russian.
• The stressed syllables are ва́ in кива́ет and ба́ in улыба́ется.

Normally, no.

Since мой друг is already the subject, adding он would usually be unnecessary. The sentence is complete as it is:

• Мой друг кивает и улыбается.

You might hear a repeated pronoun in spoken Russian for emphasis or a special conversational effect, but it is not the normal neutral version.