Sahilde yürürken sıcak kestane almadan edemiyorum.

Sahilde is sahil + -de.

• sahil = beach, shore, seaside
• -de = the locative ending, meaning in / at / on

So sahilde means on the beach or at the seaside, depending on context.

It does not mean to the beach. For to the beach, Turkish would use a different case, such as sahile.

Yürürken means while walking or when walking.

It is formed from:

• yürümek = to walk
• yürür = walks / one walks / walking in a general sense
• -ken = while / when

So yürürken literally gives the idea of while one is walking.

In this sentence, it sets the scene for the main action: while walking on the beach, the speaker cannot resist buying hot chestnuts.

Both are possible, but they are not used in exactly the same way.

• yürürken often sounds more natural for a general, repeated situation: when(ever) I walk
• yürüyorken feels more tied to a specific ongoing moment: while I am walking

In your sentence, the meaning is more like a habitual tendency, so yürürken is a very natural choice.

That -r comes from the aorist / habitual form of the verb.

• dictionary form: yürümek
• stem: yürü-
• aorist: yürür
• plus -ken: yürürken

So the word is not built directly from the infinitive yürümek. It is built from the verb form yürür.

Turkish often uses the singular noun in places where English would naturally use a plural.

So sıcak kestane can mean something like:

• hot chestnuts
• some hot chestnuts
• hot roasted chestnuts as a thing in general

This is very normal in Turkish. The plural kestaneler is possible, but it would sound more explicit and a bit less natural here.

This is a very common Turkish pattern:

V-madan edememek = to be unable to help doing something

So:

• almadan edemiyorum literally: I can't do without buying
• natural English meaning: I can't help buying

It does not mean that the speaker is physically unable to buy. It means they feel compelled to do it.

Almadan comes from:

• almak = to buy / take
• stem: al-
• -madan / -meden = without doing

So:

• almadan = without buying

By itself, almadan simply means without buying.
But together with edemiyorum, it becomes the idiomatic pattern almadan edemiyorum = I can't help buying.

This comes from etmek = to do.

In many forms, the t in etmek changes to d before a vowel-initial ending:

• ediyor
• eder
• edecek
• edemiyorum

So edemiyorum is the normal form.

Here it is built as:

• et-
• -eme- = inability / cannot
• -yor = present continuous
• -um = I

So edemiyorum means I can't do.

In Turkish, the present continuous form with -yor is often used not only for something happening right now, but also for:

• regular habits
• repeated tendencies
• personal characteristics

So edemiyorum here can mean something like:

• I can't help it
• I always end up doing it
• I just can't resist

That is perfectly natural Turkish.

Yes. It is very useful to learn V-madan edememek as a set pattern.

Examples:

• Gülmeden edemiyorum. = I can't help laughing.
• Bakmadan edemedi. = He/She couldn't help looking.
• Sormadan edemeyeceğim. = I won't be able to resist asking.

If you learn the whole pattern, it becomes much easier to understand and use sentences like this one.

Yes, Turkish word order is flexible, but the original order sounds very natural.

Sahilde yürürken sıcak kestane almadan edemiyorum flows well because it goes from:

1. place: sahilde
2. time/situation: yürürken
3. object: sıcak kestane
4. main predicate: almadan edemiyorum

You could move parts around for emphasis, but this version is smooth and idiomatic. In neutral speech, it is a very good order.