Assalamualaykum ahsan Allahu ilaykum sheikh, is the small patch under the bottom lip considered the moustache? And if so, should one trim it?

No, it is not part of the moustache. The moustache is what grows on the upper lip, and the small patch under the bottom lip is called the anfaqah, so the instruction to trim the moustache does not reach it. Whether it counts as part of the beard is a point on which the scholars differ. Some scholars held that it follows the beard and so it is not shaved, others held that it is not part of the beard, and even then said to leave it as it is unless it causes the person harm.