You can use מייצרים here. Trying to identify when you would use מפיקים, I would say, you definitely use להפיק when talking about a film, show, concert; you can use either להפיק or לייצר where you would use "extract" in English - salt, oil, wool. Otherwise use לייצר. But that's just something I thought of now, maybe there are cases that I didn't think about, maybe someone else would like to comment.
To be honest, I'm not even sure what "producing salt" means here. Does that mean a company that packages and sells table salt? Or the quarry that mines the salt? Or the lady who brings the salt containers from behind the counter and says "ta da, here's the salt"?
I think the course guys made it a bit confusing. In Hebrew if you use this sentence, it means "They are extracting/deriving our salt". In your example it would be the quarry, although salt in Israel is not mined but extracted through evaporation (AFAIK).
could this be translated as "make our salt"? Does מפיקים convey a different sense of "production"?
I have two comments about pronunciation:
Wasn't this binyan usually pronounced "mAphikim", instead of "mEphikim" shown in the audio?
Salt wasn't pronounced "mAlach", instead of "mElach", as shown in the audio? I mean, the niqqud was created exactly to preserve the Hebrew language of having pronunciations confusing the sounds of the "vocals" "a" and "e". But since the regular day-by-day Hebrew doesn't use niqqud, what would be the chance of mispronunciation?
Some sounds change when you add syllables in verbs with a diphthong merger (and some other cases).
Not sure what you are asking. Salt is melah (מֶלַח) in Hebrew, while sailor is malah (מַלָּח).