Learn Turkish: Bir şey ima edip etmediğini bilmiyorum, bu yüzden şüpheciyim. - I don't know whether he is implying something, that's why I am skeptical.

To say "whether or not" in Turkish, we use a specific pattern: [verb stem] + -(y)ip followed by [verb stem] + [negative] + -dİk. Here, the verb is etmek. So edip etmediği means "his doing and not doing". Finally, the accusative suffix -ni is added because this whole phrase is the direct object of bilmiyorum (I don't know).

Why is it just edip etmediğini instead of repeating the whole verb like ima edip ima etmediğini?

İma etmek (to imply) is a compound verb made of a noun (ima) and a helper verb (etmek). When using grammatical structures that require you to repeat the verb, you generally only repeat the helper verb. There is no need to say the noun part twice.

This lesson covers demek ki (so that means / it means that). Could we use it here instead of bu yüzden?

Not in this exact spot, because they express different logic. Bu yüzden means "for this reason" or "that's why," showing direct cause and effect (I don't know, therefore I'm skeptical). Demek ki is used for a deduction you've just made. You could flip the sentence to say: "I am skeptical, demek ki (it means that) I don't know whether he is implying something," but you cannot simply swap them.

How is the word şüpheciyim (I am skeptical) formed?