Learn Croatian: Koliko puta ste promijenili slog? - How many times did you change the typesetting?

The word koliko (how many) acts as a quantifier. Just like with numbers other than one, quantifiers like koliko, toliko, and više require the crystallized genitive form puta. You only use the base form put after the number one (jedan put) or ordinal numbers (like prvi put).

Why do we use the perfective verb promijenili here? Doesn't asking "how many times" imply a repeated action, which usually needs an imperfective verb?

This is a great catch! While it is true that repeated actions often take the imperfective, here we are counting individual, fully completed actions. Because the speaker is asking for a count of finished results, the perfective promijenili is used. If you used the imperfective mijenjali, it would sound like you were asking how many times they were in the middle of the process of changing it.

I learned that slog means "syllable". Why is it translated as "typesetting" here?

Slog is a versatile noun derived from the verb slagati (to arrange or put together). While its most common everyday meaning is indeed "syllable" (letters arranged together), in printing and publishing terminology it refers to the arrangement of text, hence "typesetting" or "layout".

What case is slog in this sentence?