There are a couple errors that are the same across all the sentences; I'll address those first, and then answer your question about the 3 different sentences.
Firstly, we make decisions, we don't take them. So instead of took the decision, your sentence should read made the decision. Additionally, you are missing a comma after if and after from me. So, the corrected version of each sentence (all of which are now perfectly grammatical):
I am curious if, when you made the decision to solicit a report entitled ‘Time as a Chimpanzee’ from me, you actually expected to receive it.
This is the most simple of the three sentences, and the most likely to actually be said. The speaker wonders if, at the time the report was asked for, the person asking for the report had a reasonable expectation of receiving the report. Very simple.
I am curious if, when you made the decision to solicit a report entitled 'Time as a Chimpanzee' from me, you actually expected to be able to receive it.
This has a slightly different meaning from the first, and probably doesn't mean what you want to say. Instead of asking if the person requesting the report expected the report to be written and handed to them, it's asking if they thought they themselves would be capable of receiving it. The implication is that they knew that something was going to happen to them that would make it impossible for them to receive the report (regardless of whether or not it was actually written, something stopped them from receiving the finished product). It's unlikely for someone to have foreknowledge of an event that will prevent them from accepting a report from somebody, so I doubt this would be said very often.
I am curious if, when you made the decision to solicit from me a report entitled 'Time as a Chimpanzee', you thought you would actually receive it.
This has the same meaning as the first sentence, with a few changes in implication. "Solicit from me a report" sounds very old-fashioned, people aren't likely to speak that way. "Solicit a report from me" is much more common. Additionally, "you thought you would actually receive it" is more rude and challenging than "you actually expected to receive it", at least in my view. With any of these sentences you're edging into dangerous territory because what you're saying is inherently a bit rude. But to me, this sentence feels even more rude than the first.
People say that.
We might get taught in schools not to say it. But to some people it's the natural way to talk. It's not ungrammatical to them.
It would probably be ungrammatical on a test of so-called standard English. But who decides what is standard? The test makers and those with socioeconomic power.
Best Answer
Question one: isn't it time, or isn't it the time?
We always say it's time, not it's the time:
We can turn all of these into questions:
And yours fits this second pattern:
It would be unidiomatic if you inserted the. People might not understand you.
In fact, the article is never used with this sense of the word, which is "the appointed, due, or proper time" (from the OED). Why not? To be honest, I have no idea. As far as I know, it's simply a matter of idiom.
Question two: acting the goat, or acting like the goat?
This is an English idiom, and you can find it in dictionaries:
You're right that acting would normally be followed by like today. But it's not ungrammatical to say it without like, and I suppose when this idiom was formed, it may have been more common to use act this way. But in any case, since it's an idiom, you shouldn't change it by inserting like.
Why do I say it's grammatical? Well, you can act silly, which shows us that act is a verb like be, become, or grow. Since it can take an adjective, we can tell these verbs take "predicative complements" (PCs) rather than objects, like most verbs. (We can tell this because objects can't be adjectives.) So just as you can say being the goat, you can say acting the goat.
Question three: the goat, or a goat?
Again, it's an idiom, so you shouldn't change it. Otherwise people won't understand!
But why the? Presumably it's the same reason we say playing the fool. We're not talking about any actual fool, nor any actual goat. We're referring to the qualities of a generic or archetypical fool or goat.
(Though, to be honest, I have no idea what makes that goat-like. It's a pretty weird idiom!)
Notes
Note one: the subordinate clause you stopped acting the goat is in the past tense, but it doesn't indicate past time. Instead, it indicates counterfactuality. In The Cambridge Grammar of the English Language, this use of the past tense is called the modal preterite.
Note two: Play the goat is a fairly unusual idiom. It was chosen, I think, to translate the original French idiom faire le zouave, which is fairly silly.