This ע has been a nightmare for me from the very beginning. I feel I'm getting better and better at guessing how to pronounce words or how to spell from hearing but I haven't spotted any pattern at all for this letter. When does it appear? How usually is it pronounced? Where? Why does "עו" instead of "או" represent /o/ here? When/where should I expect to see this letter? I'd appreciate virtually any information about it...
Most native speakers do not differentiate between the sounds that an (ע) "ayin" and an (א) "alef" make. They are basically just silent letters. In more ancient Hebrew the (ע) "ayin" would have made a more guttural sound almost like a "g." Some Hebrew speakers still differentiate. In the case you showed above the ( ו ) "vav" is what makes the "o" vowel sound.
Sadly there isn't really any trick for knowing whether it is an (א) "alef" or an (ע) "ayin." The one place where you can possibly know is on the end of a word, when on the end of a word (ע) "ayin" makes an "ah" sound and an (א) "alef" makes an "eh/ey" sound.
Apart from this, you mostly just have to memorize whether the word has an alef or an ayin. This gets easier with time, especially once you start to recognize the roots to different words.
Hope that helps...
Maghrebi, Temani, and many Mizrahi Jews would pronounce the ח, ק, ע, and צ differently, and closer to the original pronunciation. If you watch this video, you can see the man with the stubble speaks more like that: https://youtu.be/HJoW5IrN7XA Maybe it'll give you a better understanding of the phonetics! :)
From https://he.wikipedia.org/wiki/צ: the scholars disagree on whether it was like the modern pronunciation /ts/, or like the Arabic ص (IPA: /sˤ/), a heavy /s/ pronounced with the tongue making a strange position down the throat.
From https://he.wikipedia.org/wiki/ק: like the Arabic ق (IPA: /q/), similar to /k/ but with the same strange trick of the tongue down the throat.