I'm getting used to this guy's voice but I feel like he completely screwed up mặc there?
When you talk about members of your family and your relatives, you can omit "của".
E.g: Mẹ tôi = mẹ của tôi; Chú tôi = chú của tôi.
If both the noun and the pronoun are just single words, not compound words, you can omit của. For example:
Xe tôi = my car but chiếc xe của tôi.
Yes, the speaker seems to insert a nasal sound after final unvoiced stops in words in this tone, that is "một" sounds like "một n" and "mặc" like "mặc n" to me.
The curs is not consistent in the use of "the" when using classifiers: "...the purple trousers" should be valid too.
Also correct but I think they are limiting it for simplicity sake, like how they use ban for you instead of anh, chi or em.