The complete verbal phrase in the sentence is the "Disturbed by the rumor". This verbal phrase is a type called the participial phrase. This type of verbal phrase is a present or past participle, modifiers, objects or even complements. The participial phrase in the sentence is acting as an adjective to the complete thought.
They're basically teaching you/us their experiences, their way of telling you from the information they have and got from past experiences.
Every gerund has the same form - verb + -ing. So, having that in mind, the word <em>sleeping </em>is used as a gerund in this sentence, but it is also a subject of the sentence.
Answer: It looks like you are
Once they are<span> sure he is asleep, the Lilliputians, who </span>are<span> excellent mathematicians, transport </span>Gulliver<span> to the Capital</span>