Learn Hardcore Turkish: Operatör, fuar standındaki levhayı değiştirken akıllı cihaz üzerinde testler yapıyordu. - The operator was running tests on the smart device while replacing the sign at the trade show booth.

In fuar standındaki, the piece -da- is the locative case marker (meaning “at/on/in”) and -ki is a relative suffix that turns that location into an adjective modifying levha.
Morphologically:
fuar standı (fair booth) + -nda (in/on) + -ki (that is) → fuar standındaki (that is on the fair booth).
So fuar standındaki levha literally means “the board/sign that is at the fair booth.”

Why is levha marked with -yı, and why aren’t there words for “the” or “a” before operatör, levha or cihaz?

Turkish does not have articles like “the” or “a.” Definiteness of a direct object is shown by the accusative suffix -ı/-i/-u/-ü. Here, levha becomes levhayı, indicating “the sign” (a specific one).
Subjects and non‐definite nouns (like operatör or akıllı cihaz) remain unmarked.

Why is the verb form değiştirirken used, and how is it different from değişirken?

değişmek means “to change itself” (intransitive).
değiştirmek is the causative, “to change something.”
Since the operator is changing the sign (an object), we need değiştirmek.
The suffix -ken is an adverbial participle meaning “while.”
So:
değiştir- (cause to change) + -ken (while) → değiştirirken = “while (he) was changing (it).”

Why does değiştirirken not contain -yor-, while yapıyordu does?

When you form a dependent adverbial clause with -ken (meaning “while”), you do not add the continuous marker -yor- in that clause. You simply take the verb stem (here değiştir-) + -ken.
The -yor- progressive suffix only appears on the main verb. In yapıyordu, you get:
yap- (do) + -yor (progressive) + -du (past) → yapıyordu = “he was doing.”

What does üzerinde mean in akıllı cihaz üzerinde testler yapıyordu, and could we say cihazda testler yapıyordu instead?

üzerinde is a postposition meaning “on/upon.”

cihaz üzerinde test yapmak emphasizes running tests on the device’s surface or environment (e.g. software tests on the gadget).
You could also say cihazda test yapmak (“tests in/at the device”), which is common too, but it can sound more like “inside the device.”
Which one you choose depends on the nuance you want.

What tense and aspect does yapıyordu express, and how is it formed?

yapıyordu is the past continuous (he was doing). It’s built as:
yap- (root “do”)

-yor- (progressive/aspect “-ing”)
-du (past tense)
(3rd sg. zero suffix)
Vowel and consonant harmony turns yap-yor-du into yapıyordu.