The object of "I don't like people telling me what to do"?

The object of don't like in

• I don't like people telling me what to do

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

• [those] people [who are] telling me what to do,
  a complex noun phrase with a relative clause reduced by Whiz-Deletion, logically
• (∀x: PEOPLE(x)) TELL (x, I, DO (I, What))

or it could mean

• people telling me what to do, a gerund complement clause, logically
• 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

• *I don't like you tell me what to do,

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

• I don't like you to tell me what to do.

or, with the full FOR-TO complementizer,

• I don't like for you to tell me what to do.

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.