Teyzem, tavşanları görünce hemen fotoğraf çekmek istemiş; dayım da sincabı beklerken sabırlı olmuş.

They are both kinship words with -m, which means my.

• teyze = maternal aunt, specifically your mother’s sister
• dayı = maternal uncle, specifically your mother’s brother
• teyzem = my maternal aunt
• dayım = my maternal uncle

This is something English does not usually show. English just says aunt and uncle, but Turkish often distinguishes the mother’s side and father’s side.

Because this da is the separate clitic meaning also / too. In this sentence, it helps connect the second idea: my uncle too or more naturally and my uncle.

So:

• dayım da = my uncle too / and my uncle

This is different from the locative suffix -da/-de meaning in/on/at, which is written attached to the word.

• evde = at home
• dayım da = my uncle too

Because they are direct objects, and in Turkish specific/definite direct objects usually take the accusative ending.

• tavşanlar = rabbits
• tavşanları = the rabbits / those rabbits
• sincap = squirrel
• sincabı = the squirrel

A few details:

• tavşanları = tavşanlar
  • accusative -ı
• sincabı comes from sincap
  • -ı, but p changes to b before a vowel, so sincapı becomes sincabı

This often corresponds to English the, though Turkish does not have a word exactly like the.

Yes, in isolation it can.

tavşanları can mean either:

• the rabbits as a definite plural object, or
• his/her/their rabbits depending on context

In this sentence, the meaning is understood as the direct object the rabbits, because of the structure and the surrounding context.

görünce is an adverbial verb form meaning something like when seeing, when she saw, once she saw, or as soon as she saw.

It is built from:

• gör- = see
• -ünce = a form of the suffix -ınca/-ince/-unca/-ünce

So:

• tavşanları görünce = when/as soon as she saw the rabbits

This suffix creates a time relationship between two actions.

No. That is a very common confusion.

Here, görünce comes from görmek = to see, not from görünmek = to appear / to seem.

It looks similar because the suffix changes by vowel harmony:

• gelmek → gelince
• bakmak → bakınca
• görmek → görünce

So the ü is just part of the harmonized suffix here, not a different verb.

Because that is the normal Turkish expression.

Literally, çekmek often means to pull or to draw, but in this collocation:

• fotoğraf çekmek = to take a photo

Languages often use different verb combinations for the same idea. English says take a photo, but Turkish says çekmek, not almak.

Also notice the structure:

• fotoğraf çekmek istemek = to want to take a photo

After istemek (to want), the next verb usually appears in the infinitive.

Because -miş is the inferential/reported past in Turkish.

It can suggest that the speaker:

• did not directly witness it,
• is inferring it from the situation, or
• is using a storytelling/narrative tone

So:

• istedi = she wanted
• istemiş = she apparently wanted / she is said to have wanted / it seems she wanted

In many English translations, both may simply appear as wanted, but Turkish is adding extra nuance.

Yes.

• oldu = was / became
• olmuş = apparently was / turned out to be / seems to have been

So both istemiş and olmuş share that same inferential or narrative flavor.

beklerken is built with -ken, which means while or sometimes when.

It comes from:

• bekle- = wait
• bekler = waits / would wait / waiting-form base
• beklerken = while waiting

So:

• sincabı beklerken = while waiting for the squirrel

This form shows that the waiting was happening at the same time as the other situation in the clause.

Because sabırlı olmak and sabretmek are related, but not identical.

• sabırlı = patient
• sabırlı olmak = to be patient
• sabretmek = to endure patiently / to show patience

So:

• sabırlı olmuş describes his state or manner: he was patient
• sabretmiş would emphasize the act of enduring patiently

In this sentence, sabırlı olmuş focuses more on what kind of attitude he had while waiting.

Not completely. Turkish word order is flexible, but the finite verb usually stays near the end of its clause.

This sentence has a very natural order:

• Teyzem = subject
• tavşanları görünce = time clause
• hemen = adverb
• fotoğraf çekmek = infinitive phrase
• istemiş = main verb

Then:

• dayım da = subject + also
• sincabı beklerken = while-clause
• sabırlı olmuş = main predicate

You could move some parts for emphasis, for example:

• Tavşanları görünce teyzem hemen fotoğraf çekmek istemiş.

That still sounds natural. Turkish often changes word order for focus, emphasis, or style.