Learn Hardcore Turkish: Bu kanyon boyunca yürüyüş yaparak vadinin derinliklerini gözlemledik. - We observed the depths of the valley by hiking along this canyon.

boyunca is a derivational suffix/postposition meaning throughout (temporal) or along (spatial). It attaches to a noun in its bare form:
• kanyon boyunca = along the canyon
• hafta boyunca = throughout the week

Why is vadinin derinliklerini split into two case-marked nouns?

In vadinin derinliklerini:

vadinin is in the genitive case (suffix -in) to show possession (“of the valley”).
derinliklerini consists of derinlik (“depth”) + plural -ler
- accusative suffix -i (harmonized as -lerini) because it’s the specific direct object of gözlemledik (“we observed”).

What role does yürüyüş yaparak play in the sentence?

yürüyüş yapmak is a compound verb meaning to take a walk or to hike. Adding the participle suffix -arak yields yürüyüş yaparak = by taking a walk or while hiking, indicating the manner or means of observing.

Could we replace yürüyüş yaparak with yürüyerek? Is there a nuance difference?

Yes.

kanyon boyunca yürüyerek = walking along the canyon (neutral “by walking”).
kanyon boyunca yürüyüş yaparak emphasizes a planned hike or recreational activity.

Why is the main verb gözlemledik at the end of the sentence?

Turkish uses Subject-Object-Verb (SOV) word order, so modifiers and objects precede the verb. gözlemledik = we observed, with past tense marker -di and first person plural -k.

What does Bu indicate here, and why are there no articles like the or a?

Turkish has no definite/indefinite articles. Bu is a demonstrative meaning this, making Bu kanyon boyunca = along this canyon. To say a canyon, use Bir kanyon boyunca.

Can you give a word-for-word breakdown of the sentence?

Bu (this)
kanyon (canyon)
boyunca (along)
yürüyüş (walk/hike)
yaparak (by doing)
vadinin (of the valley)
derinliklerini (its depths-Accusative)
gözlemledik (we observed)