Like any other transitive verb, the subject and object of the verb like are the very same type of syntactic constituent: a noun phrase. A noun phrase can be a personal pronoun, or a noun with determiners and modifiers, or an infinitive clause, or a gerund clause. Those are all interchangeable.
The infinitive and gerund clauses are allowed to take optional subjects, but they do not require them the way the main verb does.
The object of like is the gerund clause people telling him what to do. It could have been salted peanuts instead, or himself. But it isn't. Here the object is a gerund clause. Since gerund clauses can be subjects as well as objects, you could say People telling him what to do bothers him.
In this instance, it could also have been an infinitive clause instead of a gerund clause: He doesn't like for people to tell him what to do. Using that same infinitive clause as a subject produces sentences such as For people to tell him what to do is what bothers him the most.
The rest of what you said about gerunds is incorrect because they are not special as you seem to be saying. Gerund clauses are just one possible type of noun phrase, and any noun phrase can be the verb's subject, or be the object of a verb or preposition, or be post-modified by a prepositional phrase. That's what noun phrases do.
You need to think in terms of syntactic constituents, which can be multiple words, not in terms of parts of speech of single words. A gerund clause is a noun phrase, which means it can do anything a noun phrase can do. That's how syntax works.