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That's a bit like asking why English has the letter "c" when we also have "s" and "k" - because English didn't invent the alphabet but took what the Romans gave them.
The Greeks based their alphabet on that of the Phoenecians, I believe, and so it does not fit the language perfectly.
Also, in Ancient Greek, omicron was a short sound (and a close sound) while omega was a long sound (and a more open one) -- so there the distinction made a lot of sense. Similarly with epsilon (short, close) versus eta (long, open). But alpha, iota, upsilon which could also be long and short do not have separate letters for the two quantities.
Later, the sounds changed and now omicron and omega sound the same... but Greek spelling is partly historical/etymological and so they still use both letters.
It's there but it's linked to the previous one. It's too long to be just one and, while difficult, it is possible to distinguish them (I appreciate this is particularly difficult for someone who's new to this :). It's like the name Aaron, the a's form one longer a, except in the exercise the second ο is stressed.
All the letters are neuter: το (γράμμα) άλφα, το βήτα, ... , το όμικρον... το ωμέγα. The audio file I have access to (also female voice) could be better, you're right, but we have no control over this.
You'll notice that what you see in this exercise is a pattern followed in this unit. Use that to get through it more easily, the rest of the course is a lot more enjoyable. Welcome to the course! :)