Because doubt is an old word (the OED attests it from 1225) there are a number of nuances of meaning. In summary, with my numbers not OED's:

1. A subjective state of uncertainty about the truth or reality of something;
2. (plural) A feeling of uncertainty as to something;
3. The condition of being objectively uncertain;
4. A state of affairs which gives rise to uncertainty;
5. A matter or point involved in uncertainty; a difficulty (Obsolete);
6. Apprehension, dread, fear (Obsolete);
7. A thing to be dreaded; danger, risk (Obsolete).

As a singular noun, it is decidedly abstract and refers to a state of being. Thus one can be in doubt about something. As a mass noun referring to a state of being, doubt would not normally be pluralised.

When it refers to a particular feeling of uncertainty it is pluralised. Thus if one has that feeling, one has doubts. One does not "have a doubt" because the singular doubt refers to a state of being; senses 5, 6 and 7 which refer to particular things are obsolete in Standard English.

The uses which are obsolete in Standard English may have survived or been resurrected in other dialects. The ELU references say that it does occur in Indian English.

Question one: isn't it time, or isn't it the time?

We always say it's time, not it's the time:

It's time.
It's time I went home.
It's time for you to go to bed.
It's time you started eating right.
It's time for some fun!

We can turn all of these into questions:

Isn't it time?
Isn't it time I went home?
Isn't it time for you to go to bed?
Isn't it time you started eating right?
Isn't it time for some fun?

And yours fits this second pattern:

Isn't it time you stopped acting the goat?

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:

to behave in a silly way, sometimes in order to make people laugh Insecure and lonely, he resorted to acting the goat to get people's attention.

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.