Correct. It's called nunation. When reading formal Arabic, essentially every word will have an "n" sound at the end of words. It's confusing, yes. I suggest you read up a bit on it if you're curious. But for colloquial speaking don't use nunation. For the purposes of this course they shouldnt have it in my opinion.
The sound that's being is not the ن but rather it's the sound of something called the "fathatain" that's pronounced at the end of a sentence. In the context of روزا, the sound that's being made is as if there were two dashes (one dash is called a "fatha") at the top of the letter "alif" or "ا" which is what makes that "an" sound.
This is a great question!
If you listen closely to the audio you'll hear it is pronounced more like a "u" than like an "o" in English. Sounds don't map between different languages in a 1-1 way, and in this case the "o" in English is mapping into a "u" sound, which is then represented by the letter "و".
In English, we make a distinction between "u" or "oo" as a vowel, and "w" as a consonant. However, they are closely related and have a very similar position in the mouth.
Arabic doesn't make this distinction, and instead uses "و" both in situations where we would represent the sound as a vowel, like "u" or "oo", or as a consonant, like "w".
I hope this clarifies. When learning any new writing and pronunciation system, you won't find a clean mapping between your native language and the one you're learning. Rather, you need to learn the internal logic of the new system, which is different, and recognize that proper names and borrowed words are going to be mapped into it imperfectly. If you understand the internal logic though, and how things are mapped, over time it will make sense and you'll be able to spot cognates, names, and borrowed words more easily.
As a native arabic speaker, speaking, writing and reading fluently, I find this course confusing and full of audio mistakes, I would advise you to drop this course and find another resource to learn arabic, it is difficult enough without having to relearn it the right way later on... Such a pity, it could have been a great course.