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In the sentences
I don't like people telling me what to do.
I'm fed up with you telling me what to do.
What are the objects of like and with? Is it "people" or "people telling me what to do" for the verb like?
Why can't I use a sentence just after like or with?. That is to say:
I don't like people tell me what to do.
I'm fed up with you tell me what to do.
The object of don't like in
is the noun phrase people telling me what to do. Of course it's not a single noun; it has internal structure. However, it's ambiguous. It could either mean
x: PEOPLE(x)) TELL (x, I, DO (I, What))or it could mean
TELL (People, I, DO (I, What))I will assume the second meaning. In that case, the logical structure is
NOT (LIKE (I, TELL (People, I, DO (I, What))))Since like takes a gerund complement, TELL comes out as telling in the second clause.
The last clause is a very reduced conjunctive embedded question infinitive clause, with B-Equi indicating that the subject of DO is in fact identical to the indirect object of TELL (I). Some more syntactic gymnastics is required to move what, and add complementizers and auxiliaries all around.
As to why you can't say
that's because English needs more little words and has more restrictions on verbs than Chinese. Like, for instance, needs a to complementizer in front of an infinitive object,
so what you have to say is
or, with the full FOR-TO complementizer,
Both you telling me what to do and your telling me what to do are grammatical, but they have different structures: the first is a noun phrase consisting of the head you and the modifier telling me what to do (an adjectival phrase); the second is a noun phrase consisting of the head telling me what to do and the modifier your; so telling has a different syntactic function in the two cases.
In the case of people, the form without the possessive is overwhelmingly more common, but both are still grammatical.
There has been a tradition of asserting (generally on no evidence) that the first form (without the possessive) is "wrong". This seems to have arisen from assuming that English grammar must be the same as Latin grammar.
See for example the discussion here (just the first hit I found: I'm sure there are better references).
As to your final question, most verbs cannot take a whole sentence as their object: exceptions are words of uttering:
He said "people keep telling me what to do"
where the sentence-object is generally in quotes.
Many verbs can take a whole sentence if it is within the scope of a complementiser:
I hate it when people tell me what to do
but not a bare sentence.
Your confusion comes from assuming that the construction of the two sentences is identical:
In the sentences:
I don't like people telling me what to do.
and
I'm fed up with you telling me what to do.
are of the form
(object verb)(gerund)
The problem is the form of gerund. Before identifying the type of gerund let's look at a subtle point: In the first sentence "don't like" is a verb-adverb construction that can essentially be replaced by the verb "dislike". So that the sentence becomes:
I dislike people telling me what to do.
NOTE: If the word dislike feels a bit washed down when spoken, the stress can be placed on the first syllable "dis" if there is a need to stress the "don't" in the original sentence to emphasize the emotion.
So now that we have that let's look at the respective gerunds:
I
don't likedislike people telling me what to do.I'm fed up with you telling me what to do.
In the former case it's a direct-object gerund, and in the latter there's a object of a preposition gerund (the preposition being with and object being you).
These phrases are grammatically correct.
The rest of what I was going to say about the construction of your replacements is nicely said by John Lawler :-)
The objects are "you telling me what to do" and "people telling me what to do", not "people" or "you".
Technically, the grammar of your initial examples is incorrect. It's not that you dislike the person; it's that you dislike their action. And that action is "telling me what to do". So the phrase "telling me what to do" is a gerund, a phrase that is built on a verb but acts as a noun.
Given that, the correct forms for your initial sentences are:
I don't like people's telling me what to do.
I'm fed up with your telling me what to do.
In this case, I concede that the first sentence, though grammatically correct, sounds awkward. So I recommend re-wording it as follows:
I don't like when people tell me what to do.
I'm fed up with your telling me what to do.
Your final examples are grammatically incorrect and sound awkward to a native English speaker. The reason they are incorrect is that the words "like" and "with" require an object. And the phrases "people tell me what to do" and "you tell me what do to" cannot act as the object because they are complete sentences in their own right.
