Kuzenim rujunu tazeledikten sonra hafif parfüm sürdü ve aynaya gülümsedi.

Kuzen means cousin. The ending -im is the 1st person singular possessive suffix, so kuzenim means my cousin.

Turkish often shows possession by adding a suffix to the noun, instead of using a separate word like my.

Rujunu can be broken down like this:

• ruj = lipstick
• -u = his/her/its possessive suffix
• -nu = accusative marking after a possessed noun

So rujunu means her/his lipstick as a specific object.

In this sentence, the most natural interpretation is her lipstick:
Kuzenim rujunu tazeledi = My cousin refreshed her lipstick.

Not 100% from grammar alone. Grammatically, rujunu just means his/her lipstick. But in this sentence, the most natural reading is that the cousin refreshed her own lipstick.

Turkish often leaves this kind of thing to context.

Tazeledikten sonra means after refreshing or after having refreshed.

It comes from:

• tazele- = to refresh, renew, touch up
• -dikten sonra = after doing

So:

• rujunu tazeledikten sonra = after refreshing her lipstick

This pattern is very common in Turkish:

• yemek yedikten sonra = after eating
• eve geldikten sonra = after coming home

Because Turkish often drops subject pronouns when they are already clear.

Here, kuzenim already tells us who the subject is, so Turkish does not need to add o (he/she). This is very normal.

Because parfüm is being used as an indefinite/non-specific direct object.

In Turkish:

• a specific/definite direct object usually takes the accusative
• a non-specific direct object often stays in the bare form

So:

• parfüm sürdü = applied perfume
• parfümü sürdü = applied the perfume / a specific perfume

Also, hafif means light, mild, or subtle, and adjectives normally come before the noun in Turkish.

Yes. Hafif bir parfüm sürdü would also be natural and would mean something like she applied a light perfume.

Without bir, hafif parfüm sürdü still sounds natural and keeps the object more general.

Yes. Sürmek has several meanings.

Two common ones are:

• to drive
• to apply / spread on

In this sentence, it has the second meaning. Turkish uses sürmek for applying things like:

• cream
• ointment
• lipstick
• perfume/cologne

So parfüm sürdü means applied perfume or put on perfume.

Because gülümsemek here takes the dative case when you smile at someone or something.

• ayna = mirror
• ayna-ya = to/at the mirror

So:

• aynaya gülümsedi = she smiled at the mirror

The -y- is a buffer consonant added because ayna ends in a vowel.

Gülümsemek means to smile.
Gülmek usually means to laugh, though in some contexts it can also mean to smile.

Here gülümsedi is more exact, because the action is specifically smiling.

A natural breakdown is:

• Kuzenim = my cousin
• rujunu tazeledikten sonra = after refreshing her lipstick
• hafif parfüm sürdü = applied light perfume
• ve aynaya gülümsedi = and smiled at the mirror

Turkish often puts the time/action background before the main action, and the verb usually comes near the end of its clause. So the structure may feel different from English, but it is very typical Turkish.

Yes. Both sürdü and gülümsedi have the same subject: kuzenim.

So the sentence means that my cousin did both actions:

• applied perfume
• smiled at the mirror

Turkish does not need to repeat the subject when it stays the same.