Amcamla dayım bahçede otururken yağmur başladı, yine de içeri girmek istemediler.

Yes, both are usually translated as uncle, but Turkish makes a family distinction that English usually does not:

• amca = your father’s brother
• dayı = your mother’s brother

So amcamla dayım refers to two different kinds of uncles.

In Turkish, each kinship noun normally keeps its own possessive ending.

• amcam = my paternal uncle
• dayım = my maternal uncle

So amcamla dayım is natural Turkish for my uncle and my other uncle. English can often say my uncle and aunt with my only once, but Turkish usually repeats the possession on each family term.

Here it works like and.

The suffix -la / -le originally means with, but when it links two nouns, it often functions like and:

• annemle babam = my mother and father
• amcamla dayım = my paternal uncle and my maternal uncle

So in this sentence, amcamla dayım is a compound subject, not really with my uncle.

Otururken comes from oturmak + -ken.

• oturmak = to sit
• -ken = while, when

So bahçede otururken means while sitting in the garden or while they were sitting in the garden.

In this sentence, it gives background action: they were sitting in the garden, and during that time the rain started.

Because otururken is the normal, compact, natural way to say while sitting / while they were sitting.

Turkish often uses verb + -ken without spelling out all the tense/person detail that English would use. The subject is understood from context.

Forms like oturuyorken or oturuyorlarken can exist, but they sound heavier and are less common here. In everyday Turkish, otururken is the most natural choice.

By default, the subject of a -ken clause is understood from context, and here it matches the main subject: amcamla dayım.

So the sentence means that my two uncles were the ones sitting in the garden when the rain started.

Turkish often leaves this kind of subject unstated when it is already clear.

No, they are different.

• bahçe-de: the -de is a suffix meaning in / at / on
  • bahçede = in the garden
• yine de: de is a separate word here, part of the expression yine de
  • yine de = nevertheless / even so / still

A useful spelling clue:

• case suffixes attach to the word: bahçede
• separate particles are written separately: yine de

Turkish does not need a dummy subject like English it.

So instead of it started raining, Turkish can simply say:

• yağmur başladı = the rain started

You can also hear:

• yağmur yağmaya başladı = it started to rain

Both are correct. Yağmur başladı is shorter and very natural.

Yine de adds the idea of despite that or even so.

So the logic is:

• it started raining
• nevertheless, they still did not want to go inside

It is stronger than just ama in this context.
Compare:

• ama = but
• yine de = still / even so / nevertheless

So yine de highlights that their reaction was a bit unexpected.

İçeri girmek is the normal idiomatic way to say to go inside / to enter.

• içeri = inside, inward
• girmek = to enter

So içeri girmek is very natural.

You could also hear içeriye girmek, but içeri without the extra ending is extremely common.
İçeriye gitmek is possible in some contexts, but girmek fits better when the idea is actually entering a space.

Because içeri often behaves like a directional adverb, not like a regular object noun.

So Turkish commonly says:

• içeri girmek = go in
• dışarı çıkmak = go out
• yukarı çıkmak = go up

You may also hear forms like içeriye, but the shorter adverbial form içeri is very common and natural.

The verb is plural because the subject is plural: amcamla dayım refers to two people.

• istemediler = they did not want

The ending -ler shows plural. Turkish often does not use a separate pronoun like onlar if the subject is already clear from the nouns or from the verb ending.

So saying just istemediler is completely normal.

Yes, Turkish word order is fairly flexible, although this version sounds natural and neutral.

This sentence has a very typical flow:

• subject: Amcamla dayım
• place/background: bahçede otururken
• event: yağmur başladı
• contrast: yine de
• action they refused: içeri girmek istemediler

Some parts could move for emphasis, but the original order is smooth and standard. Turkish often moves elements around to highlight what is most important, while English is usually more fixed.