Learn Turkish: Türkçesi gayet iyi. - His Turkish is perfectly good.

Gayet means "perfectly," "quite," or "highly," and it usually carries an approving or reassuring tone. While çok iyi just means "very good," gayet iyi implies his language skills are "perfectly fine" or "completely satisfactory."

Why is there an -si suffix on Türkçesi?

The -(s)i is the 3rd person singular possessive suffix. When talking about someone's specific ability or command of a language, Turkish treats the language as a possessed noun. So Türkçesi literally translates to "his/her Turkish."

Could we use the o kadar ... ki structure from this lesson with this sentence?

Yes, but you would have to expand the sentence. If you swap gayet for o kadar, it becomes Türkçesi o kadar iyi ki... (His Turkish is so good that...). Because o kadar sets up an expectation for a result, you would then need to add a clause like ...herkes onu Türk sanıyor (...that everyone thinks he is Turkish).

There is no verb in this sentence. Should there be a -dir suffix at the end?

You could say gayet iyidir, but in everyday conversation, the third-person "to be" suffix (-dir / -dır) is usually dropped when stating simple facts or personal observations. Leaving it as iyi is grammatically complete and sounds much more natural here.