Mekanizma yavaş çalışırken bile tüm fonksiyonları yerine getirebiliyordu.

The suffix -ken attaches to a verb stem to form a temporal participle meaning “while …‐ing.” Here, çalış- (to work) + -ken = çalışırken, i.e. “while it was working.”

Both yavaş and yavaşça can function as adverbs meaning “slowly,” but in everyday Turkish yavaş is more common with machinery or processes. yavaş çalışmak (“to work slowly”) feels more idiomatic than yavaşça çalışmak in this context.

bile is a particle meaning “even.” Placed after a phrase, it adds emphasis of exception:
çalışırken bile = “even while it was working.”
It underlines that despite working slowly, the mechanism still performed all functions.

Turkish marks definite direct objects with the accusative suffix. Here tüm (all) makes fonksiyonlar (functions) specific, so we add -ı → fonksiyonları to show it’s the object of yerine getirmek.

• yerine – adverbial phrase “in place”
• getir- – verb stem “to bring/execute”
• -ebil- – ability suffix “can/be able to”
• -iyor- – continuous-tense marker (“-ing”)
• -du – past-tense marker (“was”)
• (3rd person singular is unmarked)
Together, yerine getir-ebil-iyor-du = “it was able to perform (carry out).”

In Turkish, person and number are encoded in the verb ending. Here getirebiliyordu is 3rd person singular, so the subject (the mechanism) is understood and doesn’t need a pronoun.

Yes. tüm and bütün both mean “all” or “whole.” There’s no real difference in this sentence, though bütün can sound slightly more formal or emphatic.
– tüm fonksiyonları = bütün fonksiyonları