Learn English – third person singular verbs in questions

From usingenglish.com...

The finite forms of a verb are the forms where the verb shows tense, person or singular plural.
Non-finite verb forms have no person, tense or number.

finite I go, she goes, he went.
non-finite To go, going (the infinitive, and the gerund/present participle).

Thus in OP's first three examples, coughs, had compromised, will have gone are "finite" forms of the verbs to cough, to compromise, to go. In the second and third cases, the modal auxiliary verbs will and had/have are the only ways in English to indicate verb tense, aspect, and modality, apart from the endings -s (present tense, third person singular), -ed (past tense), and -ing ("continuous" aspect).

In OP's last example, were is a "tensed" (finite) form indicating past action and plural subject (or second person singular "you"). So those verb forms change for present activity or singular subject, but the "non-finite" present participle talking doesn't...

The girls were talking to John. (past action, plural subject)
The girls are talking to us. (present action, plural subject)
John was talking to me. (past action, singular subject)
John is talking to you. (present action, singular subject)

Also note that it makes no difference whether the object (John, us, me, you) is singular or plural.

Typically, only the first verb in each clause can be finite:

I play.
I played.
She plays.
She played.

As you can see, this verb changes form. It shows tense (present or past) and agrees with the subject (adding -s if the subject is third person singular present). This must be our finite verb.

Our example is a declarative clause. To make it into a question, we should turn it into an interrogative clause. And for this sort of sentence, we do that by swapping the subject and auxiliary verb, like this:

She was safe.　　→　　Was she safe?

This is called Subject-Auxiliary Inversion (SAI). We use SAI to mark these sentences as questions. And in this case, SAI is possible because we have the auxiliary be.

But what if our example doesn't have an auxiliary verb? Let's look at our example from earlier:

I play.
I played.
She plays.
She played.

We can't swap the subject and auxiliary verb if we don't have one! Let's solve that problem by adding the meaningless ("dummy") auxiliary do:

I do play.
I did play.
She does play.
She did play.

Now we've got two verbs! And only the first can be finite. In this case, that means do! And as you can see, do now changes form, but play doesn't! That's because play is no longer finite.

Now we can make it into a question. So, let's do Subject-Auxiliary Inversion again, moving the auxiliary before the subject I:

Do I play?
Did I play?
Does she play?
Did she play?

These questions are formed correctly. But what about your example?

*Does she plays?

It doesn't work! Why not? Because we'd have two finite verbs.

In this answer, the * symbol marks a sentence as ungrammatical, and bold in examples marks a verb as finite.